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 whi…