Genetically Modified T Cells in Treating Patients With Stage III-IV Non-small Cell Lung Cancer or Mesothelioma

Inclusion Criteria:

* ELIGIBILITY FOR ENROLLMENT/SCREENING (ARMS 1 AND 2): Histopathological documentation of NSCLC or mesothelioma
* ELIGIBILITY FOR ENROLLMENT/SCREENING (ARMS 1 AND 2): Patients must be able to give informed consent
* ELIGIBILITY FOR ENROLLMENT/SCREENING (ARMS 1 AND 2): Patients must be able to provide blood and tumor samples and undergo the procedures required for this protocol
* Arm 2 ONLY: Surgically operable NSCLC or mesothelioma
* ELIGIBILITY FOR TREATMENT ON ARM 1: Patients must express human leukocyte antigen (HLA)-A\*0201
* ELIGIBILITY FOR TREATMENT ON ARM 1: Evidence of WT1 tumor expression
* ELIGIBILITY FOR TREATMENT ON ARM 1: Patients must have received at least one line of therapy for NSCLC or mesothelioma or previously documented to have declined therapy
* ELIGIBILITY FOR TREATMENT ON ARM 1: NSCLC patients with a mutation in epidermal growth factor receptor (EGFR) or anaplastic lymphoma kinase (ALK) must have demonstrated progression or intolerance to at least one of the corresponding targeted therapies (for example erlotinib or crizotinib)
* ELIGIBILITY FOR TREATMENT ON ARM 1: Bi-dimensionally measurable disease by palpation, clinical exam, or radiographic imaging (X-ray, computed tomography \[CT\] scan, positron emission tomography \[PET\] scan, magnetic resonance imaging \[MRI\], or ultrasound)
* ELIGIBILITY FOR TREATMENT ON ARM 1: Ninety days must have passed since the last doses of radiation or chemoradiation treatment involving lung tissue or thorax prior to T cell infusion (to avoid confounding pneumonitis)
* ELIGIBILITY FOR TREATMENT ON ARM 1: Patients treated with prior immunotherapy including and not limited to vaccines, cytokines, T cell stimulating agents, cytotoxic T lymphocyte antigen 4 (CTLA4) inhibitors and programmed death (PD)-1 check point inhibitors are allowed on therapy provided they did not have any severe grade 4 toxicities due to prior therapy and any toxicities due to prior therapy should have resolved, if resolvable to less than or equal to grade 1
* ELIGIBILITY FOR TREATMENT ON ARM 2: Patients must express HLA-A\*0201
* ELIGIBILITY FOR TREATMENT ON ARM 2: Evidence of WT1 tumor expression
* ELIGIBILITY FOR TREATMENT ON ARM 2: Ninety days must have passed since the last definitive doses of radiation or chemoradiation treatment prior to T cell infusion (to avoid confounding pneumonitis)

Exclusion Criteria:

* EXCLUSION FOR ENROLLMENT/SCREENING (ARMS 1 AND 2)
* Eastern Cooperative Oncology Group (ECOG) performance status \>= 2
* Active autoimmune disease (e.g., systemic lupus erythematosus, vasculitis, infiltrating lung disease, inflammatory bowel disease) in which possible progression during treatment would be considered unacceptable by the investigators
* Any condition or organ toxicity deemed by the principal investigator (PI) or the attending physician to place the patient at unacceptable risk for treatment on the protocol
* Men or women of reproductive ability who are unwilling to use effective contraception or abstinence; women of childbearing potential must have a negative urine pregnancy test within 2 weeks prior to first infusion
* Pregnant women and nursing mothers will be eligible for screening only to test HLA type by saliva or buccal swab and WT1 expression from previously collected tissue sample
* Clinically significant and ongoing immune suppression including, but not limited to, systemic immunosuppressive agents such as cyclosporine or corticosteroids, chronic lymphocytic leukemia (CLL), uncontrolled human immunodeficiency virus (HIV) infection, or solid organ transplantation
* EXCLUSION FOR TREATMENT (ARMS 1 AND 2)
* Exclusions for the leukapheresis procedure (this can be performed at a later time of symptoms resolve):

  * Infection, with or without antibiotic treatment
  * Recent hepatitis exposure (hepatitis B or C antigenemia)
  * Pregnancy or nursing
  * HIV or human T-lymphotropic virus (HTLV) infection
  * Positive result on standard test for syphilis (STS)
* Unable to generate antigen-specific WT1-specific CD8+ T cells for infusions; however, the patient will have the option to receive WT1-specific T-cells if a lower than planned number of cells is available
* Documented infections or known oral temperature \> 38.2 degrees Celsius (C) fewer than 72 hours prior to receiving study treatment or systemic infection requiring chronic maintenance; the start of treatment may be delayed
* Systemic steroids should be stopped 2 weeks before the start of treatment; topical and inhaled steroids are allowed
* Untreated central nervous system (CNS) metastasis that are \> 1 cm or symptomatic are not allowed; (patients with CNS metastases \> 1 cm or symptomatic that have been treated and demonstrated to be radiologically and clinically stable for at least 4 weeks are allowed)
* White blood cells (WBC) \< 2,000/ul
* Hemoglobin (Hb) \< 8 g/dL
* Absolute neutrophil count (ANC) \< 1,000/ul
* Platelets \< 50,000/ul
* New York Heart Association functional class III-IV heart failure, symptomatic pericardial effusion, stable or unstable angina, symptoms of coronary artery disease (CAD), congestive heart failure, clinically significant hypotension or history of an ejection fraction of =\< 30% (echocardiogram or multi-gated acquisition scan \[MUGA\])
* Clinically significant pulmonary dysfunction, as determined by medical history and physical exam; patients so identified will undergo pulmonary functions testing and those with forced expiratory volume in 1 second (FEV1) \< 2.0 L or diffusion capacity of the lungs for carbon monoxide (DLCO) (corrected for Hb) \< 50% will be excluded
* Creatinine \> 1.5 x the upper limit of normal
* Aspartate aminotransferase/alanine aminotransferase (AST/ALT) \> 5 x upper limits of normal (ULN)
* Bilirubin \> 3 x ULN that cannot be attributed to NSCLC metastasis
* HIV or HTLV infection