Modified T Cells, Chemotherapy, and Aldesleukin With or Without LV305 and CMB305 in Treating Participants With Advanced or Recurrent Sarcoma

Inclusion Criteria:

* Histopathologic documentation of synovial sarcoma or myxoid liposarcoma with the diagnosis of advanced or recurrent disease who have received prior standard chemotherapy. Patients with other sarcoma subtypes if proven to be NY-ESO-1 positive and meeting all other eligibility criteria listed below will also be included.
* Tumor expression of NY-ESO-1 (2+ staining or \> 25%) by immunohistochemistry (IHC).
* Expression of HLA-A\*0201.
* Eastern Cooperative Oncology Group (ECOG)/ Zubrod performance status of '0-1'
* Life expectancy \> 6 months.
* Electrocardiography (ECG) without evidence of clinically significant arrhythmia or ischemia.
* Women of childbearing potential (WOCBP) must be using at least one highly effective or two effective accepted methods of contraception to avoid conception throughout the study in such a manner that the risk of pregnancy is minimized. Suggested precautions should be used to minimize the risk or pregnancy for at least 1 month before start of therapy, and while women are on study for up to 3 months after T cell infusion and/or at least 3 months after the study agents LV305 or CMB305 are stopped. WOCBP include any female who has experienced menarche and who has not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation or bilateral oophorectomy) or is not postmenopausal.
* Men must be willing and able to use an acceptable method of birth control such as latex condom during the dosing period and for at least 3 months after completion of the study agent administration (T cell infusion and/or LV305 or CMB305) if their sexual partners are WOCBP.
* Willing and able to give informed consent.
* (Prior to treatment) Note: evaluate at least 1 week before T cell infusion. a. Adequate venous access - consider peripherally inserted central catheter (PICC) or central line. b. ECOG/Zubrod performance status of '0-1. c. Bi-dimensionally measurable disease by palpation on clinical exam, or radiographic imaging (X-ray, computed tomography \[CT scan\]). d. At least 4 weeks must have elapsed since the last chemotherapy, immunotherapy, radiotherapy or major surgery. At least 6 weeks for nitrosoureas, mitomycin C and liposomal doxorubicin. e. Toxicity related to prior therapy must either have returned to =\< grade 1, baseline, or been deemed irreversible. f. Persons of reproductive potential must agree to use and utilize an adequate method of contraception throughout treatment and for at least 3 months after study drug is stopped. g. Willing and able to give informed consent.

Exclusion Criteria:

* Patients with active infections or oral temperature \> 38.2 Celsius (C) within 72 hours of leukapheresis. The procedure may be deferred.
* Investigational therapy within 3 weeks.
* Prior administration of other NY-ESO-1 targeting immunotherapeutics.
* Significant immunosuppression from concurrent, recent (=\< 4 weeks ago) or anticipated treatment with systemic corticosteroids at any dose, or other immunosuppressive medications such as methotrexate, cyclosporine, azathioprine (antihistamines, non-steroidal anti-inflammatory drugs and aspirin permitted) or conditions such as common variable hypogammaglobulinemia or exposures such as large field radiotherapy.
* Cancer therapies, including chemotherapy, radiation, biologic, or kinase inhibitors, granulocyte colony-stimulating factor (G-CSF), granulocyte-macrophage colony-stimulating factor (GM-CSF) within 3 weeks prior.
* Psychiatric, other medical illness or other condition that in the opinion of the principal investigator (PI) prevents compliance with study procedures or ability to provide valid informed consent.
* Significant autoimmune disease with the exception of alopecia, vitiligo, hypothyroidism or other conditions that have never been clinically active or were transient and have completely resolved and require no ongoing therapy.
* Myocardial infarction within 6 months of study initiation, active cardiac ischemia or New York Heart Association (NYHA) grade III or IV heart failure.
* Peripheral blood leukocyte count (white blood cells \[WBC\]) \< 3000/mm\^3.
* Absolute neutrophil count =\< 1500/mm\^3.
* Platelets \< 75000/mm\^3.
* Hemoglobin \< 10 gm/dL.
* Alanine aminotransferase (ALT), and aspartate aminotransferase (AST) \> 2.5 x upper limit of normal (ULN).
* Total serum bilirubin \> 1.5 x ULN (patients with Gilbert's disease may be included if their total bilirubin is =\< 3.0 mg/dL).
* Creatinine \> 1.5 x ULN. If higher check 24hr clearance, if \< 50 ml/min then patient will be excluded.
* INR (prothrombin time ratio) or partial thromboplastin time (PTT) \> 1.5 x ULN (Please note: patients with hematopoietic cell transplantation (Hct) \< 30%, WBC \< 2500/mm/\^3 and platelets \< 50,000/mm\^3 immediately prior to leukapheresis. The procedure may be deferred.)
* History of other cancer within 3 years (except non-melanoma cutaneous malignancies and cervical carcinoma in situ).
* Positive screening tests for human immunodeficiency virus (HIV), hepatitis (Hep) B, Hep C, active tuberculosis or recent (\< 2 week ago) clinically significant infection or evidence of active HIV, Hep B, or Hep C. (Note: If positive results are not indicative of true active or chronic infection, the patient can be treated.)
* Brain metastases considered unstable as: a. without confirmed stability over 60 days in patients previously treated with prior surgery or radiation; OR b. associated with symptoms and/or findings; OR c. requiring corticosteroids or anticonvulsants in the prior 60 days.
* Pregnant, planning to become pregnant, or breast feeding.
* Known allergy(ies) to any component of CMB305 or LV305.
* Men or women of reproductive ability who are unwilling to use effective contraception and women of childbearing potential who are unwilling to undergo pregnancy testing before and during the study.
* 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 carbon monoxide diffusing capability (DLco) (correlation for hemoglobin \[corr for Hgb\]) \< 75% will be excluded.
* Significant cardiovascular abnormalities as defined by any one of the following: a. congestive heart failure, b. clinically significant hypotension, c. symptoms of coronary artery disease, d. presence of cardiac arrhythmias on electrocardiography (EKG) requiring drug therapy, e. ejection fraction \< 50 % (dobutimine stress echo).
* Active and untreated central nervous system (CNS) metastasis.
* Autoimmune disease: patients with a history of inflammatory bowel disease are excluded from this study, as are patients with a history of autoimmune disease (e.g. systemic lupus erythematosus, vasculitis, infiltrating lung disease) whose possible progression during treatment would be considered by the investigator to be unacceptable.
* Steroids are not permitted 3 days prior to T cell infusion and concurrently during therapy.
* No prisoners or children will be enrolled on this study.