Trastuzumab and Combination Chemotherapy in Treating Patients With Locally Recurrent or Metastatic Urinary Tract Cancer

Inclusion Criteria: * Histologic diagnosis of urothelial carcinoma (TCC or Squamous) that is either metastatic or locally recurrent and not curable by surgery or radiation therapy; patients must have HER2 overexpression as documented by ANY of the following: 1. 2+ or 3+ staining by immunohistochemistry, or 2. a positive FISH score defined as \> 2 with the Vysis system or \> 4 with the Ventana system, or 3. an elevated serum HER2 of \> 16 ng/ml using the OSDI assay; please note: * Tissue from either the primary or metastatic site must be tested for HER2 status determination * All patients must have a blood sample drawn for HER2 serologic testing * If the available tissue is from the primary tumor and is HER2 negative and if the serum is negative, to qualify for the study a biopsy of a metastatic site should be done and the patient will be eligible ONLY if this demonstrates HER2 over-expression * All sites and measurements of disease must be clearly documented in the pre-study forms * All prior local or adjuvant systemic therapy including the type of chemotherapy must be clearly documented in the pre-study form Note: Patients with Her-2 negative tumors are not eligible for treatment on this protocol but their response to other therapy and survival will also be evaluated * Bidimensionally measurable or evaluable disease not previously radiated * No prior systemic chemotherapy for metastatic disease; patients may have received adjuvant chemotherapy if completed at least 6 months prior to beginning this protocol treatment; type and number of courses of prior chemotherapy must be clearly documented * A performance status of 0-2 by Southwest Oncology Group criteria and a life expectancy of greater than 12 weeks * Serum creatinine =\< 2.0 mg% * Granulocyte count \>= 1,500/mm\^3 * Platelet \>= 100,000/mm\^3 * Total bilirubin =\< 1.5 mg/dl * No significant cardiac disease and must have adequate cardiac function (ejection fraction \>= 50% or higher than the lower limit of institutional normal) as determined by a MUGA scan or 2-D echocardiogram within 4 weeks from registration, and no evidence of symptomatic coronary artery disease (baseline EKG must show no active ischemia); patients must not have history of congestive heart failure * If patients have received prior radiation therapy, disease must be present outside of radiated fields and at least 4 weeks must have elapsed since discontinuation of that therapy; the nadirs of RT leukopenia and thrombocytopenia must be surpassed with evidence of hematologic recovery * No prior malignancy is allowed, except for adequately treated basal cell (or squamous cell) skin cancer, in situ carcinoma of any site or other cancer for which the patient is currently disease free * Patient may not have unresolved bacterial infection * Pregnant or lactating women may not participate; this is to avoid potential harm since the effects of study drugs on the fetus or the nursing infant are not known; women/men of reproductive potential may not participate unless they have agreed to use an effective contraceptive method * HIV -positive patients may not participate; this is to avoid additional complications that immune suppression and HIV infection may cause due to the intense nature of the chemotherapy in this trial * Timing guidelines for pre-study labs (excluding HER2 determination) and measurements; * To be completed within 14 days prior to registration; pre-study labs required for determination of eligibility * To be completed within 28 days prior to registration: x-rays, scans or physical examination used for tumor * All patients must be informed of the investigational nature of this study and must sign an informed consent in accordance with institutional and federal guidelines * All patients must be registered with the UM Cancer Center Clinical Trials Office at 734-647-8174 (Joanne Goodson) prior to instituting therapy