Durvalumab and Standard Chemotherapy Before Surgery in Treating Patients With Variant Histology Bladder Cancer

Inclusion Criteria:

* Signed informed consent.
* Eastern Collaborative Oncology Group (ECOG) performance status score of 0 or 1.
* Body weight \> 30 kg.
* Absolute neutrophil count (ANC) \>= 1500 mm\^3 (within 28 days before the first study treatment).
* Hemoglobin \>= 9.0 g/dL (within 28 days before the first study treatment).
* Platelet count \>= 100,000 per mm\^3 (within 28 days before the first study treatment).
* Serum bilirubin =\< 1.5 X upper limit of normal (ULN) (within 28 days before the first study treatment). Subjects with Gilbert's syndrome will be considered after consultation with the principal investigator (PI).
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 X ULN (within 28 days before the first study treatment).
* For subjects who will be treated with dose dense methotrexate, vinblastine, doxorubicin, and cisplatin (DD MVAC) or cisplatin and gemcitabine (CG), creatinine clearance \>= 50 mL/min as measured based on Cockcroft-Gault glomerular filtration rate estimation (within 28 days before the first study treatment).
* For subjects who will be treated with carboplatin and gemcitabine (Carbo Gem), creatinine clearance \>= 30 mL/min as measured based on Cockcroft-Gault glomerular filtration rate estimation (within 28 days before the first study treatment).
* Anticipated life expectancy of \>= 12 weeks as assessed by the investigator.
* Histologically proven carcinoma of the bladder of variant urothelial carcinoma histologies which include squamous, adenocarcinoma, nested, plasmacytoid, micropapillary, glandular differentiation, lipid cell, clear cell, undifferentiated, giant cell, trophoblastic, sarcomatoid, carcinosarcoma; subjects with mixed cell types are eligible.
* Clinical T stage 2 (cT2) T4a, N0 N1, M0 disease. Clinical T stage is based on the transurethral resection of bladder tumor (TURBT) sample and imaging studies. Subjects must undergo cystoscopy and TURBT as part of screening within 30 days prior to registration.
* Abdominal/pelvic imaging by computed tomography (CT) or magnetic resonance imaging (MRI) scan; chest imaging by CT scan or x-ray (CT/positron emission tomography (PET) within 30 days prior to registration.
* Resting 12 lead electrocardiogram (ECG) documenting Fridericia's correction formula (QTcF) =\< 470 ms.
* Consent to provide a formalin fixed paraffin embedded (FFPE) tissue block and 1 hematoxylin and eosin (H and E) slide (preferred) or one of the following:

  * 10 unstained slides and 1 H and E slide OR
  * Tissue block punches and 1 H and E slide, OR
  * 4 to 6 cores and 1 H and E slide.
* Willing and able to comply with the protocol for the duration of the study including undergoing treatment, scheduled visits and examinations including follow up.
* For female subjects:

  * Women \< 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of exogenous hormonal treatments and if they have luteinizing hormone and follicle stimulating hormone levels in the post-menopausal range for the institution or underwent surgical sterilization (bilateral oophorectomy or hysterectomy).
  * Women \>= 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of all exogenous hormonal treatments, had radiation induced menopause with last menses \> 1 year ago, had chemotherapy induced menopause with last menses \> 1 year ago, or underwent surgical sterilization (bilateral oophorectomy, bilateral salpingectomy or hysterectomy).

Exclusion Criteria:

* Prior treatment with systemic cytotoxic chemotherapy for muscle invasive bladder cancer (MIBC).
* Class III or IV heart failure, according to New York Heart Association Classifications. For patient on the dd MVAC or Cis-Gem arm, left ventricular ejection fraction of less than 50%
* Administration of an investigational therapeutic agent within 28 days of protocol registration.
* Current participation in a trial using an investigational agent. Subjects may participate in non-interventional, observational studies.
* Prior treatment with an anti-programmed cell death 1(PD1) or anti--programmed cell death ligand 1(PDL1) inhibitor including durvalumab.
* Receiving chronic systemic steroid therapy in dosing exceeding 10 mg daily of prednisone or equivalent per day within 7 days prior to the first dose of study treatment.
* History of another malignancy within 5 years before the first dose of study drug, or any evidence of residual disease from a previously diagnosed malignancy. Subjects with the following are allowed on study:

  * Adequately treated non melanoma skin cancer or lentigo maligna without evidence of disease
  * Adequately treated carcinoma in situ without evidence of disease e.g., cervical cancer in situ.
* Immunosuppressive medication within 28 days before the first dose of durvalumab, with the exceptions of intranasal and inhaled corticosteroids or systemic corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone, or an equivalent corticosteroid. The following are exceptions to this criterion:

  * Intranasal, inhaled, topical steroids, or local steroid injections (e.g., intra articular injection)
  * Systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or its equivalent
  * Steroids as premedication for hypersensitivity reactions (e.g., CT scan premedication) or anti emetic during chemotherapy.
* History of allogenic organ transplantation.
* Active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease \[e.g., colitis or Crohn's disease\]), diverticulitis (with the exception of diverticulosis), systemic lupus erythematosus, sarcoidosis syndrome, or Wegener syndrome (granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis, hypophysitis, uveitis, etc). The following are exceptions to this criterion:

  * Subjects with vitiligo or alopecia
  * Subjects with hypothyroidism (e.g., following Hashimoto syndrome) stable on hormone replacement
  * Any chronic skin condition that does not require systemic therapy
  * Subjects with celiac disease controlled by diet alone.
* Uncontrolled intercurrent illness, including but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, interstitial lung disease, serious chronic gastrointestinal conditions associated with diarrhea, or psychiatric illness/social situations that would limit compliance with study requirement, substantially increase risk of incurring adverse events (AEs) or compromise the ability of the subject to give written informed consent.
* History of active primary immunodeficiency.
* Active infection including:

  * Tuberculosis (clinical evaluation that includes clinical history, physical examination and radiographic findings, and tuberculosis (TB) testing in line with local practice)
  * Hepatitis B (known positive hepatitis B virus (HBV) surface antigen (HBsAg) result)
  * Hepatitis C
  * Human immunodeficiency virus (HIV) (positive HIV 1/2 antibodies).
  * Subjects with a past or resolved HBV infection (defined as the presence of hepatitis B core antibody (anti HBc) and absence of HBsAg) are eligible. Subjects positive for hepatitis C (HCV) antibody are eligible only if polymerase chain reaction is negative for HCV ribonucleic acid (RNA).
* Receipt of live attenuated vaccine within 30 days prior to the first dose of study medication. Note: subjects, if enrolled, should not receive live vaccine while receiving study medication and up to 30 days after the last dose of study medication.
* Pregnant or lactating.
* Male or female subject of reproductive potential who are not willing to employ effective birth control from screening to 90 days after the last dose of durvalumab monotherapy.
* Known allergy or hypersensitivity to any of the study medications or any of the study medication excipients.
* Judgment by the investigator that the subject is unsuitable to participate in the study and the subject is unlikely to comply with study procedures, restrictions and requirements.