Cabozantinib S-malate and Nivolumab With or Without Ipilimumab in Treating Patients With Metastatic Genitourinary Tumors

Inclusion Criteria: * Patients in the phase I portion must have: * Histologically confirmed diagnosis of metastatic, genitourinary solid tumor * Metastatic disease defined as new or progressive lesions on cross-sectional imaging; patients must have at least: * One evaluable site of disease * Or, appearance of one new bone lesion * Patients in the expansion portion must have: * Histologically confirmed diagnosis of metastatic: * Urothelial carcinoma of the bladder, urethra, ureter, renal pelvis, OR * Clear cell renal cell carcinoma OR * Adenocarcinoma of the bladder OR * Non-resectable squamous cell carcinoma of the penis OR * Squamous or small cell carcinoma of the bladder, renal medullary carcinoma (RMC), sarcomatoid bladder and renal cell carcinomas, plasmacytoid carcinoma of the bladder or other rare bladder/kidney cancer histology AND * Patients with urothelial cancer or renal cell carcinoma must have progressive metastatic disease defined as new or progressive lesions on cross-sectional imaging; patients must have at least: * One measurable site of disease (according to RECIST criteria) or bone disease by NaF PET/CT * Patients must have either progressed on at least one standard therapy or there must be no standard treatment that has been shown to prolong survival for the patient's disease (patients with urothelial carcinoma who are cisplatin-ineligible may receive protocol therapy as a first line therapy); patients may have received any number of prior cytotoxic agents * Age \>= 18 years on day of consent. Because no dosing or adverse event data are currently available on the use of cabozantinib in combination with nivolumab and ipilimumab in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Karnofsky performance status \>= 70% * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,200/mcL * Platelets \>= 75,000/mcL * Total bilirubin =\< 1.5 x upper limit of normal (ULN); for subjects with known Gilbert's disease or similar syndrome with slow conjugation of bilirubin, total bilirubin =\< 3.0 mg/dL * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional upper limit of normal (ULN) * Creatinine =\< 1.5 x ULN OR creatinine clearance \>= 40 mL/min/1.73 m\^2 (calculated using the Chronic Kidney Disease-Epidemiology Collaboration \[CKD-EPI\] equation or Cockcroft-Gault formula) for patients with creatinine levels above institutional normal * Hemoglobin \>= 9 g/dL * Serum albumin \>= 2.8 g/dL * Lipase and amylase =\< 2.0 x ULN and no radiologic or clinical evidence of pancreatitis * Urine protein/creatinine ratio (UPCR) =\< 2 * Serum phosphorus \>= lower limit of normal (LLN) (if below LLN, for asymptomatic patients replacement may be initiated if clinically indicated without delaying the start of study treatment) * Serum calcium \>= LLN (if below LLN, for asymptomatic patients replacement may be initiated if clinically indicated without delaying the start of study treatment) * Serum magnesium \>= LLN (if below LLN, for asymptomatic patients replacement may be initiated if clinically indicated without delaying the start of study treatment) * Serum potassium \>= LLN (if below LLN, for asymptomatic patients replacement may be initiated if clinically indicated without delaying the start of study treatment) * Women of childbearing potential must have a negative pregnancy test at screening; women of childbearing potential include women who have experienced menarche and who have not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or are not postmenopausal; postmenopause is defined as amenorrhea \>= 12 consecutive months; Note: women who have been amenorrheic for 12 or more months are still considered to be of childbearing potential if the amenorrhea is possibly due to prior chemotherapy, antiestrogens, ovarian suppression or any other reversible reason * The effects of the drugs used in this trial on the developing human fetus are unknown; however, cabozantinib was embryolethal in rats at exposures below the 140 mg dose in the label, with increased incidences of skeletal variations in rats and visceral variations and malformations in rabbits; for this reason and because tyrosine kinase inhibitors agents as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception, as defined below, prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and for 7 months after completion of all study medications; women treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and for 5 months after completion of all study medications * Sexually active subjects (men and women) must agree to use medically accepted barrier methods of contraception (e.g., male or female condom) during the course of the study and for 5 or 7 months for women or men respectively, after the last dose of study drugs, even if oral contraceptives are also used; all subjects of reproductive potential must agree to use both a barrier method and a second method of birth control during the course of the study and for 5 or 7 months for women and men respectively after the last dose of study drugs * Tissue availability for PD-L1 expression is mandatory for enrollment; however if archived tissue is unavailable the patient will be given the option to consent to pre and post treatment tissue biopsies; tissue biopsies will be collected pretreatment (prior to the first dose of therapy) and post treatment (after at least 1 dose, preferably 2 doses of nivolumab) * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * The subject has received cytotoxic chemotherapy (including investigational cytotoxic chemotherapy) or biologic agents (e.g., cytokines or antibodies) within 3 weeks, or nitrosoureas/mitomycin C within 6 weeks before the first dose of study treatment * Patients who have been previously treated with MET or VEGFR inhibitors (except for patients on renal cell cancer \[RCC\] cohort) are not eligible for the expansion cohorts but can enroll in the phase I portion * Prior treatment with any therapy on the PD-1/PD-L1 axis or CTLA-4 inhibitors unless enrolling the urothelial carcinoma with previous checkpoint inhibition therapy expansion cohort * The subject has received radiation therapy: * To the thoracic cavity or abdomen within 3 months before the first dose of study treatment, or has ongoing complications, or is without complete recovery and healing from prior radiation therapy * To bone or brain metastasis within 3 weeks before the first dose of study treatment * To any other site(s) within 28 days before the first dose of study treatment * The subject has received radionuclide treatment within 6 weeks of the first dose of study treatment * The subject has received prior treatment with a small molecule kinase inhibitor within 14 days or five half-lives of the compound or active metabolites, whichever is longer, before the first dose of study treatment * The subject has received prior treatment with hormonal therapy within 14 days or five half-lives of the compound or active metabolites, whichever is longer, before the first dose of study treatment; subjects receiving gonadotropin-releasing hormone (GnRH) agonists and antagonists are allowed to participate * The subject has received any other type of investigational agent within 28 days before the first dose of study treatment * The subject has not recovered to baseline or Common Terminology Criteria for Adverse Events (CTCAE) =\< grade 1 from toxicity due to all prior therapies except alopecia and other non-clinically significant adverse events (AEs) defined as lab elevation with no associated symptoms or sequelae * The subject has active brain metastases or epidural disease; subjects with brain metastases previously treated with whole brain radiation or radiosurgery or subjects with epidural disease previously treated with radiation or surgery who are asymptomatic and do not require steroid treatment for at least 2 weeks before starting study treatment are eligible; neurosurgical resection of brain metastases or brain biopsy is permitted if completed at least 3 months before starting study treatment; baseline brain imaging with contrast-enhanced CT or magnetic resonance imaging (MRI) scans for subjects with known brain metastases is required to confirm eligibility * The subject has prothrombin time (PT)/international normalized ratio (INR) or partial thromboplastin time (PTT) test \>= 1.5 x the laboratory ULN within 7 days before the first dose of study treatment * No concomitant treatment with warfarin is permitted. Aspirin (up to 325 mg/day), thrombin or factor Xa inhibitors, low-dose warfarin (=\<1 mg/day), prophylactic and therapeutic low molecular weight heparin (LMWH) are permitted * The subject requires chronic concomitant treatment of strong CYP3A4 inducers (e.g., dexamethasone, phenytoin, carbamazepine, rifampin, rifabutin, rifapentine, phenobarbital, and St. John's wort); Because the lists of these agents are constantly changing, it is important to regularly consult medical reference texts such as the Physicians' Desk Reference may also provide this information; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * The subject has experienced any of the following: * Clinically-significant gastrointestinal bleeding within 6 months before the first dose of study treatment * Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood within 3 months before the first dose of study treatment * Any other signs indicative of pulmonary hemorrhage within 3 months before the first dose of study treatment * The subject has tumor invading any major blood vessels * The subject has evidence of tumor invading the gastrointestinal (GI) tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or endobronchial tumor within 28 days before the first dose of cabozantinib * The subject has uncontrolled, significant intercurrent or recent illness including, but not limited to, the following conditions: * Cardiovascular disorders including: * Congestive heart failure (CHF): New York Heart Association (NYHA) class III (moderate) or class IV (severe) at the time of screening * Concurrent uncontrolled hypertension defined as sustained blood pressure (BP) \> 140 mmHg systolic, or \> 90 mmHg diastolic despite optimal antihypertensive treatment within 7 days of the first dose of study treatment * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \> 500 ms within 28 days before randomization; Note: if initial QTcF is found to be \> 500 ms, two additional electrocardiograms (EKGs) separated by at least 3 minutes should be performed; if the average of these three consecutive results for QTcF is =\< 500 ms, the subject meets eligibility in this regard * Any history of congenital long QT syndrome * Any of the following within 6 months before the first dose of study treatment: * Unstable angina pectoris * Clinically-significant cardiac arrhythmias * Stroke (including transient ischemic attack \[TIA\], or other ischemic event) * Myocardial infarction * Cardiomyopathy * Gastrointestinal disorders particularly those associated with a high risk of perforation or fistula formation including: * Any of the following that have not resolved within 28 days before the first dose of study treatment * Intra-abdominal tumor/metastases invading GI mucosa * Active peptic ulcer disease * Diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Malabsorption syndrome * Any of the following within 6 months before the first dose of study treatment: * Abdominal fistula * Gastrointestinal perforation * Bowel obstruction or gastric outlet obstruction * Intra-abdominal abscess; Note: complete resolution of an intra-abdominal abscess must be confirmed prior to initiating treatment with cabozantinib even if the abscess occurred more than 6 months before the first dose of study treatment * Other disorders associated with a high risk of fistula formation including percutaneous endoscopic gastrostomy (PEG) tube placement within 3 months before the first dose of study therapy * Other clinically significant disorders such as: * Severe active infection requiring systemic treatment within 14 days before the first dose of study treatment * Serious non-healing wound/ulcer/bone fracture within 28 days before the first dose of study treatment * History of organ transplant * Concurrent uncompensated hypothyroidism or thyroid dysfunction within 7 days before the first dose of study treatment (for asymptomatic patients with an elevated thyroid stimulating hormone \[TSH\], thyroid replacement may be initiated if clinically indicated without delaying the start of study treatment) * History of major surgery as follows: * Major surgery within 3 months of the first dose of cabozantinib; however, if there were no wound healing complications, patients with rapidly growing aggressive cancers, may start as soon as 6 weeks if wound has completely healed post-surgery * Minor surgery within 1 month of the first dose of cabozantinib if there were no wound healing complications or within 3 months of the first dose of cabozantinib if there were wound complications excluding core biopsies and Mediport placement * In addition, complete wound healing from prior surgery must be confirmed at least 28 days before the first dose of cabozantinib irrespective of the time from surgery * The subject is unable to swallow tablets * History of severe hypersensitivity reaction to any monoclonal antibody * The subject is unable or unwilling to abide by the study protocol or cooperate fully with the investigator or designee * For disease specific studies: the subject has had evidence within 2 years of the start of study treatment of another malignancy which required systemic treatment * History of allergic reactions attributed to compounds of similar chemical or biologic composition to cabozantinib, nivolumab, ipilimumab or other agents used in study * Pregnant women are excluded from this study because cabozantinib is a tyrosine kinase inhibitor with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with cabozantinib, breastfeeding should be discontinued if the mother is treated with cabozantinib; these potential risks may also apply to other agents used in this study * Human immunodeficiency virus (HIV)-positive patients are eligible if on stable dose of highly active antiretroviral therapy (HAART), CD4 counts are greater than 350 and viral load is undetectable * Patients are excluded if they have a positive test for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection * Patients with active autoimmune disease or history of autoimmune disease that might recur, which may affect vital organ function or require immune suppressive treatment including systemic corticosteroids, should be excluded; these include but are not limited to patients with a history of immune related neurologic disease, multiple sclerosis, autoimmune (demyelinating) neuropathy, Guillain-Barre syndrome, myasthenia gravis; systemic autoimmune disease such as systemic lupus erythematosus (SLE), connective tissue diseases, scleroderma, inflammatory bowel disease (IBD), Crohn's, ulcerative colitis, hepatitis; and patients with a history of toxic epidermal necrolysis (TEN), Stevens-Johnson syndrome, or phospholipid syndrome should be excluded because of the risk of recurrence or exacerbation of disease; patients with vitiligo, endocrine deficiencies including thyroiditis managed with replacement hormones including physiologic corticosteroids are eligible; patients with rheumatoid arthritis and other arthropathies, Sjogren's syndrome and psoriasis controlled with topical medication and patients with positive serology, such as antinuclear antibodies (ANA), anti-thyroid antibodies should be evaluated for the presence of target organ involvement and potential need for systemic treatment but should otherwise be eligible * Patients are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger (precipitating event) * Patients are excluded if they have a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration; inhaled or topical steroids and adrenal replacement doses \< 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease; patients are permitted to use topical, ocular, intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic absorption); physiologic replacement doses of systemic corticosteroids are permitted, even if \< 10 mg/day prednisone equivalents; a brief course of corticosteroids for prophylaxis (e.g., contrast dye allergy) or for treatment of non-autoimmune conditions (e.g., delayed-type hypersensitivity reaction caused by contact allergen) is permitted * Patients who have had evidence of active or acute diverticulitis, intra-abdominal abscess, GI obstruction and abdominal carcinomatosis which are known risk factors for bowel perforation should be evaluated for the potential need for additional treatment before coming on study