Testing the Addition of Immunotherapy Before Surgery for Patients With Sarcomatoid Mesothelioma

This phase II trial evaluates the safety and effectiveness of giving immunotherapy (nivolumab and ipilimumab) before surgery for controlling disease in patients with stage I-IIIa sarcomatoid mesothelioma. Immunotherapy with monoclonal antibodies, such as nivolumab and ipilimumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving immunotherapy before surgery may be more effective at controlling disease in patients with sarcomatoid mesothelioma than giving immunotherapy alone.

Inclusion Criteria: * Sarcomatoid or sarcomatoid-dominant (\> 50%) biphasic, pleural mesothelioma * Stage: I-IIIA disease per Union for International Cancer Control (UICC) TNM Classification of Malignant Tumours 8th edition * Measurable disease or non-measurable disease as defined * No prior treatment which would be considered treatment for the primary neoplasm or impact the primary endpoint * No treatment with hormones or other chemotherapeutic agents except for hormones administered for non-disease-related conditions (e.g., insulin for diabetes and or hormonal therapy for breast, prostate cancer etc.) * Not pregnant and not nursing, because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown \* Therefore, for women of childbearing potential only, a negative pregnancy test done =\< 14 days prior to registration is required * Age \>= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 or Karnofsky \>= 60% * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 * Leukocytes \>= 2,000/mm\^3 * Platelet count \>= 100,000/mm\^3 * Creatinine =\< 1.5 x upper limit of normal (ULN) OR creatinine clearance \>= 40 mL/min * Total bilirubin =\= 60% * STEP 2 ELIGIBILITY CRITERIA: Predicted forced expiratory volume in 1 second (FEV1) \> 35% and postoperative predicted diffusion capacity of the lung for carbon monoxide (DLCO) \> 35% * STEP 2 ELIGIBILITY CRITERIA: Registration to step 2 no less than 21 days and no more than 90 days after the last dose of neoadjuvant therapy Exclusion Criteria: * No patients deemed to be unresectable or poor surgical candidates * No patients with chest wall invasion, peritoneal spread, contralateral pleural involvement, mediastinal organ involvement, vertebral involvement, or metastases to contralateral intrathoracic lymph nodes, or any supraclavicular nodes * No patients with a history of symptomatic interstitial lung disease