Nivolumab, Cabozantinib S-Malate, and Ipilimumab in Treating Patients With Recurrent Stage IV Non-small Cell Lung Cancer

Inclusion Criteria (Step 0): * Patients with tumors with the following molecular alterations must submit testing results via Medidata Rave to determine eligibility to Arm T; the study chair, co-chair, biology co-chair, or a delegate must review the molecular testing and agree that the testing meets one of the molecular eligibility criteria below: * ROS1 gene rearrangement by fluorescence in situ hybridization (FISH) or deoxyribonucleic acid (DNA) analysis (may have progressed on prior crizotinib therapy) * MET exon 14 splice mutations on DNA analysis (may have progressed on prior crizotinib therapy) * MET high amplification by FISH or DNA analysis or other MET mutations predicted to be sensitive to MET inhibitor (no prior targeted therapy allowed) * RET gene rearrangement by FISH or DNA analysis (no prior targeted therapy allowed) * Institutions will be notified of the patient's eligibility status for Arm T within two (2) business days of submission of the molecular testing reports * If patients do not have tumors with the above molecular alterations noted proceed directly to step 1 Inclusion Criteria (Step 1): * For patients with known molecular alterations, institution has been notified that patient is deemed eligible for Arm T per review of molecular testing reports * Pathologically confirmed non-squamous non-small cell lung carcinoma (NSCLC) * Stage IV disease (includes M1a, M1b, or recurrent disease), according to the 7th edition of the lung cancer tumor, node, and metastasis (TNM) classification system * Predominant non-squamous histology (patients with NSCLC not otherwise specified \[NOS\] are eligible); mixed tumors will be categorized by the predominant cell type; if small cell elements are present the patient is ineligible * Tumors must be tested and known negative for EGFR tyrosine kinase inhibitor (TKI) sensitizing mutations (EGFR exon 19 deletions, L858R, L861Q, G719X) and ALK gene rearrangements by routine Clinical Laboratory Improvement Act (CLIA)-certified clinical testing methods; negative circulating tumor DNA results alone are not acceptable; prior testing for tumor PD-L1 status is not required * Patients must have progressed radiographically following first line platinum-based chemotherapy, no additional lines of therapy are permitted * NOTE: Prior adjuvant chemotherapy for early stage disease does not count as one line of therapy if 12 months or greater elapsed between completion of adjuvant therapy and initiation of first-line systemic therapy; if less than 12 months elapsed, adjuvant chemotherapy counts as one line of therapy * Exception for targeted therapy sub-study (Arm T): At least one line of prior chemotherapy or targeted therapy is required, but there is no limit on number of prior treatments * Patients must have measurable disease as defined by RECIST v. 1.1 criteria; baseline measurements and evaluation of all sites of disease must be obtained within 4 weeks prior to registration * Any prior chemotherapy (based on administration schedule) must have been completed in greater than or equal to the following times prior to registration: * Chemotherapy/ targeted oral therapy administered in a daily or weekly schedule must be completed \>= 1 week prior to registration; * Any chemotherapy administered in an every 2 week or greater schedule must be completed \>= 2 weeks prior to registration * Additionally, patients should be recovered to equal to or less than grade 1 toxicities related to any prior treatment, unless adverse event (AE)(s) are clinically nonsignificant and/or stable on supportive therapy * Patients with no known brain metastasis must have baseline brain imaging within 12 weeks prior to study registration not demonstrating brain metastases OR patients with known brain metastases must have baseline brain imaging within 4 weeks prior to study registration and meet all of the following criteria: * Have completed treatment to all symptomatic brain metastases (with whole brain radiation or radiosurgery) \>= 4 weeks prior to registration, or have undergone complete neurosurgical resection \>= 3 months prior to registration * Be clinically stable from brain metastases at time of screening, if no treatment was administered * Known leptomeningeal disease is not allowed * Patients must have Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Patients must have anticipated life expectancy greater than 3 months * Absolute neutrophil count \>= 1,500/mm\^3 (within 2 weeks prior to registration) * Platelets \>= 100,000/mm\^3 (within 2 weeks prior to registration) * Hemoglobin \>= 9 g/dL (within 2 weeks prior to registration) * Subject has prothrombin time (PT)/international normalized ratio (INR) and partial thromboplastin time (PTT) test =\< 1.3 x the laboratory upper limit of normal (ULN) (within 2 weeks prior to registration) * Total bilirubin =\< 1.5 x ULN (within 2 weeks prior to registration) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 3 x ULN (within 2 weeks prior to registration) * Serum albumin \>= 2.8 g/dL (within 2 weeks prior to registration) * Serum calcium (absolute or albumin corrected), magnesium and potassium \>= lower limit of normal (LLN) (within 2 weeks prior to registration) * NOTE: serum calcium, magnesium and potassium can be replaced if values are below LLN * Creatinine =\< 1.5 x ULN or calculated (Cockcroft-Gault formula) or measured creatinine clearance \>= 50 mL/min/1.73 m\^2 (normalized to body surface area \[BSA\]) for patients with creatinine levels greater than 1.5 times the institutional normal (within 2 weeks prior to registration) * Screening urine dipstick must equal 0 or "trace"; if urine dipstick results are \>= 1+, or if dipstick was not performed, calculation of urine protein creatinine (UPC) is required and patients must have a UPC ratio =\< 1 to participate in the study (within 2 weeks prior to registration) * Patients must have corrected QT interval calculated by the Fridericia formula (QTcF) =\< 500 ms within 28 days before registration * Patients must be able to swallow tablets * All females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy * A female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Women of childbearing potential (WOCBP) and males who are sexually active with WOCBP must use an accepted and effective method of contraception or abstain from sexual intercourse for at least one week prior to the start of treatment, and continue for 5 months after the last dose of protocol treatment for women of childbearing potential and 7 months after the last dose of protocol treatment for males who are sexually active with WOCBP Exclusion Criteria (Step 1): * Small cell elements are present * Prior anti-MET therapy such as crizotinib or cabozantinib, or PD-1/PD-L1 immune checkpoint inhibitor therapy (such as nivolumab, pembrolizumab, atezolizumab) or CTLA4 inhibitor therapy (such as ipilimumab); prior allergic reaction to small molecule tyrosine kinase inhibitors or monoclonal antibodies * Exception for targeted therapy sub-study (Arm T): Prior crizotinib may be allowed depending on the gene alteration * Prior radiation therapy for bone metastasis within 2 weeks; any other radiation therapy within 4 weeks prior to registration * Known leptomeningeal disease * Impaired decision-making capacity (IDMC) * Clinically-significant gastrointestinal bleeding within 6 months prior to registration * Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood within 3 months prior to registration * Drug induced pneumonitis within 3 months prior to registration * Signs indicative of pulmonary hemorrhage within 3 months before the first dose of study treatment * Radiographic evidence of cavitating pulmonary lesion(s) * Tumor invading any major blood vessels * Evidence of tumor invading the GI tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or mainstem endobronchial tumor within 28 days before the first dose of cabozantinib * Concomitant anticoagulation with oral anticoagulants (e.g., warfarin, direct thrombin and Factor Xa inhibitors) or platelet inhibitors (e.g., clopidogrel); allowed anticoagulants are the following: * Low-dose aspirin for cardioprotection (per local applicable guidelines) is permitted * Low molecular weight heparins (LMWH) or unfractionated heparin is permitted * Anticoagulation with therapeutic doses of LMWH is allowed in subjects without known brain metastases who are on a stable dose of LMWH for at least 6 weeks before first dose of study treatment, and who have had no clinically significant hemorrhagic complications from the anticoagulation regimen or the tumor * Concomitant treatment of strong CYP3A4 inducers (e.g., dexamethasone, phenytoin, carbamazepine, rifampin, rifabutin, rifapentine, phenobarbital, and St. John's wort) * 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) \> 150 mm Hg systolic, or \> 100 mm Hg diastolic despite optimal antihypertensive treatment within 7 days prior to registration * Any of the following within 6 months prior to registration: * Unstable angina pectoris * Clinically-significant cardiac arrhythmias * Stroke (including transient ischemic attack \[TIA\], or other ischemic event) * Myocardial infarction * Gastrointestinal disorders associated with a high risk of perforation or fistula formation within 3 months prior to registration: * Active peptic ulcer disease * Inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Known malabsorption syndrome * Bowel obstruction or gastric outlet obstruction * Percutaneous endoscopic gastrostomy (PEG) tube placement * Gastrointestinal disorders associated with a high risk of perforation or fistula formation within 6 months prior to registration: * Abdominal fistula * Gastrointestinal perforation * 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 prior to registration * Any of the following conditions: * Grade 3 or greater infection, or infection requiring intravenous systemic treatment within 28 days prior to registration; patients should be off antibiotics at the time of registration. * Serious non-healing wound/ulcer/bone fracture within 28 days prior to registration * History of organ transplant * Concurrent symptomatic untreated hypothyroidism within 7 days prior to registration * History of surgery as follows: * Major surgery (as an example, surgery requiring anesthesia and a \> 24 hour hospital stay) within 3 months prior to registration, with wound healing at least 28 days prior to registration * Minor surgery within 28 days prior to registration with complete wound healing at least 10 days prior to registration * Minor procedures within 7 days prior to registration such as thoracentesis, paracentesis, or 18 g or smaller needle biopsy of tumor * Patients with clinically relevant ongoing complications from prior surgery are not eligible * Currently active other malignancies which require systemic treatment * 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 * Active autoimmune disease or known history of autoimmune disease for which recurrence may affect vital organ function or require immune suppressive treatment including systemic corticosteroids; 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, autoimmune hepatitis; 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 type I diabetes mellitus or 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 * Ongoing major illness or psychosocial issues that would limit compliance with the protocol * Pregnant or breast-feeding * Patients with known human immunodeficiency virus (HIV) disease taking antiretroviral therapy are excluded because there are no safety data with the combination of antiretroviral therapy and cabozantinib or ipilimumab or nivolumab with ipilimumab * Patients with known chronic active hepatitis B (defined as a positive hepatitis B surface antigen and/or hepatitis B viral load in the last