Nab-Paclitaxel and Bevacizumab or Ipilimumab as First-Line Therapy in Treating Patients With Stage IV Melanoma That Cannot Be Removed by Surgery

Inclusion Criteria: * Histologic or cytologic proof of surgically unresectable stage IV malignant melanoma - including that of uveal and mucosal origin * Note: biopsy can be of locoregional disease in setting of clinically evident stage IV disease; a biopsy of the primary tumor alone does not fulfill this requirement * No more than 2 prior courses of systemic therapy for metastatic melanoma * For patients with metastatic melanoma not of uveal origin, v-raf murine sarcoma viral oncogene homolog B1 (BRAF) V600 mutation determination using a Clinical Laboratory Improvement Amendments (CLIA)-approved testing method on metastatic tumor tissue * NOTE: patients with metastatic melanoma of uveal origin do not need to have formal BRAF testing due to low probability of a BRAF V600 mutation in their metastatic tumor * Measurable disease; note: disease that is measurable by physical examination only is not eligible * Life expectancy of \>= 4 months * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Absolute neutrophil count \>=1500/mL (obtained =\< 14 days prior to registration/randomization) * Platelet count \>= 100,000 x 10\^9/L (obtained =\< 14 days prior to registration/randomization) * Hemoglobin \>= 9 g/dL (obtained =\< 14 days prior to registration/randomization) (patients may be transfused to meet this requirement) * Creatinine =\< 1.5 x upper limit of normal (ULN) (obtained =\< 14 days prior to registration/randomization); institutional norms are acceptable * Total bilirubin =\< 1.5 mg/dL (obtained =\< 14 days prior to registration/randomization) (exception: patients with documented Gilbert?s syndrome are allowed to participate despite elevated bilirubin) * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 2.5 x ULN and serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 2.5 x ULN (obtained =\< 14 days prior to registration/randomization) * Alkaline phosphatase =\< 2.5 x ULN (obtained =\< 14 days prior to registration/randomization); if bone metastasis is present in the absence of liver metastasis then =\< 5 x ULN * Urine dipstick for proteinuria \< 2+ (obtained =\< 14 days prior to registration/randomization) (patients discovered to have \>= 2+ proteinuria on dipstick urinalysis at baseline should undergo a 24 hour urine collection and must demonstrate =\< 1 g of protein in 24 hours to be eligible) * Negative serum pregnancy test done =\< 7 days prior to registration/randomization, for women of childbearing potential only * Note: * Females: adequate contraception must be used by both patient and partner while receiving study drug and for 12 weeks after the last dose of study drug * Males: adequate contraception must be used by both patient and partner while receiving study drug; men who have a partner of childbearing age should also avoid fathering a child for 6 months after the last dose of study drug * Ability to understand and the willingness to sign a written informed consent document * Mayo Rochester patients only: willingness to provide mandatory blood samples for research purposes Exclusion Criteria: * Brain metastases per magnetic resonance imaging (MRI) or computed tomography (CT) * Note: patients who have had therapy for brain metastasis (i.e., surgical resection, whole brain radiation, or stereotactic radiosurgery \[SRS\] even if stable) are not eligible * Other investigational agents =\< 4 weeks prior to registration/ randomization * Anti-cancer therapy (including immunotherapy) =\< 4 weeks prior to registration/randomization; exception: adjuvant Leukine =\< 14 days prior to registration/randomization * Prior treatment in the adjuvant or metastatic setting with any of the following: * Agents disrupting VEGF activity or targeting vascular endothelial growth factor receptor (VEGFR); * Ipilimumab; * Or taxane based chemotherapy regimens (including paclitaxel, docetaxel, cabazitaxel or nab-paclitaxel) * Major surgical procedure, open biopsy, or significant traumatic injury =\< 4 weeks prior to registration/randomization; (port-a-cath placement does not count as a major surgical procedure and patients can be enrolled at any time after placement) * Fine needle aspirations or core biopsies =\< 7 days prior to registration/ randomization * Planned/or anticipated major surgical procedure during the course of the study * Other medical conditions including but not limited to: * History of liver disease such as cirrhosis, chronic active hepatitis, chronic persistent hepatitis or hepatitis B or C * Active infection requiring parenteral antibiotics * Poorly controlled high blood pressure (\>= 150 mmHg systolic and/or 100 mmHg diastolic) despite treatment * New York Heart Association class II-IV congestive heart failure * Serious cardiac arrhythmia requiring medication * Myocardial infarction or unstable angina =\< 6 months prior to registration/randomization * Clinically significant peripheral vascular disease * Deep venous thrombosis or pulmonary embolus =\< 1 year of registration/randomization * Ongoing need for full-dose oral or parenteral anticoagulation * Ongoing anti-platelet treatment other than low-dose aspirin (i.e., aspirin 81 mg by mouth daily) * Active bleeding or pathological conditions that carry high risk of bleeding (e.g., known esophageal varices, etc.) * Serious, non-healing wound (including wounds healing by secondary intention), ulcer or bone fracture * History of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess =\< 6 months prior to registration/randomization * History of central nervous system (CNS) disease (e.g., vascular abnormalities, etc.), clinically significant stroke or transient ischemic attack (TIA) =\< 6 months prior to registration/randomization, seizures not controlled with standard medical therapy * Radiographically documented tumor invading major blood vessels * History of hypertensive crisis or hypertensive encephalopathy * Any of the following: * Pregnant women * Nursing women * Men and women of reproductive potential who are not using effective birth control methods Note: women of childbearing potential must have a negative serum pregnancy test =\< 7 days prior to registration/randomization; adequate contraception must be used while receiving study drug and for 12 weeks after the last dose of study drug, by both women and men and by both patient and partner; men who have a partner of childbearing potential should also avoid fathering a child for 6 months after the last dose of study drug * Existence of peripheral sensory neuropathy \>= grade 2 (from any cause) * History of other malignancy =\< 5 years with the exception of basal cell or squamous cell carcinoma of the skin, treated with local resection only, or carcinoma in situ (e.g. of the cervix, breast, prostate, etc.) * Radiation therapy (other than palliative) =\< 2 weeks prior to randomization; note: patients who have had \> 25% of their functional bone marrow irradiated are not eligible for this trial * Active or recent history of hemoptysis (\>= 1/2 teaspoon of bright red blood per episode) =\< 30 days prior to registration/randomization * Known hypersensitivity to any of the components of ipilimumab, bevacizumab, or nab-paclitaxel * History of inflammatory bowel disease (e.g., Crohn?s, ulcerative colitis) - note patients with irritable bowel syndrome are eligible * Diagnosis of autoimmune disease (i.e., rheumatoid arthritis, scleroderma, systemic lupus erythematosus \[SLE\], autoimmune vasculitis, Guillain-Barre syndrome, etc.), regardless if patient is currently receiving treatment at time of registration/randomization * Systemic corticosteroids use =\< 2 weeks, regardless of indication; note: patients who are on inhaled corticosteroids are eligible