Testing the Addition of Paclitaxel Administered Into the Abdominal Cavity Combined With Chemotherapy for Patients With Gastric Cancer Spread to the Abdominal Cavity

STEP 0 REGISTRATION: * Patient must be at least 18 years of age * Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Patient must have histologically or cytologically confirmed microsatellite stable (MSS) or mismatch repair (MMR) protein expression proficient primary gastric or gastroesophageal adenocarcinoma (Siewert 3) with synchronous cytology positive disease (cyt+) OR peritoneal carcinomatosis detected by imaging, laparoscopy or laparotomy. Patients with microsatellite instability-high (MSI-H/dMMR) mismatch repair deficient disease are not eligible * Patient must have received a minimum of 3 months and a maximum of 6 months of first line systemic treatment * Patient must be registered to Step 0 within 4 weeks of the last dose of first line systemic therapy. Patient must not have any ongoing significant adverse events that would prohibit them from undergoing a diagnostic laparoscopy procedure followed by further systemic and intraperitoneal therapy * Patient must have no evidence of small or large bowel obstruction other than gastric outlet obstruction due to primary malignancy * Patient must have no evidence of solid organ metastases except for ovarian metastases. Baseline imaging must be done within 30 days prior to Step 0 registration * Patient must have no evidence of clinically significant radiologic peritoneal disease progression during first line systemic therapy * Patient must have no evidence of extensive retroperitoneal lymph node metastases not amenable to resection during gastrectomy * Patient must have no history of prior surgery that would preclude safe diagnostic laparoscopy and port placement * Patient must have no evidence of massive ascites on imaging or history of two therapeutic paracentesis with drainage of more than 1.0 liter of ascites each time in 30 days prior to Step 0 registration * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better * Patient must not have any uncontrolled intercurrent illness or any other significant condition(s) that would make this protocol unreasonably hazardous * Patient must not have any known contraindications or drug allergies to the protocol treatment agents: paclitaxel, 5-fluorouracil, or leucovorin * Leukocytes ≥ 2,000/uL (≤ 30 days prior to Step 0 registration) * Absolute neutrophil count (ANC) ≥ 1,500/uL (≤ 30 days prior to Step 0 registration) * Platelets ≥ 75,000/uL (≤ 30 days prior to Step 0 registration) * Total bilirubin ≤ 1.5 institutional upper limit of normal (ULN). If patient has Gilbert's syndrome, total bilirubin must be \< 2.0 mg/dL (≤ 30 days prior to Step 0 registration) * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT)(serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3.0 x institutional ULN (≤ 30 days prior to Step 0 registration) * Creatinine clearance ≥ 30 mL/min (estimated using Cockcroft and Gault formula or measured) (≤ 30 days prior to Step 0 registration) * Hemoglobin ≥ 8 g/dL (≤ 30 days prior to Step 0 registration) * Serum albumin ≥ 2.5 g/dL (≤ 30 days prior to Step 0 registration) * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months of Step 0 registration are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Patient must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used * All patients of childbearing potential must have a blood test or urine study within 14 days prior to Step 0 registration to rule out pregnancy * A patient of childbearing potential is defined as anyone, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Patient must not expect to conceive or father children by using accepted and effective method(s) of contraception (or by abstaining from sexual intercourse) for the duration of their participation in the study. Arm A patients must adhere to the contraceptive requirements outlined in the product specific package inserts while on protocol treatment. Arm B patients must continue contraceptive measures for at least 3 months after the last dose of protocol treatment. In addition, both Arm A and Arm B patients who continue with targeted agents must adhere to the contraceptive requirements outlined in the product specific package inserts while on protocol treatment * Patient must have the ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver and/or family member available will also be considered eligible STEP 1 RANDOMIZATION: * Patient must have undergone a diagnostic laparoscopy with peritoneal lavage performed and aspiration for cytology obtained * The extent of peritoneal disease burden must have been assessed during the diagnostic laparoscopy with the Peritoneal Cancer Index (PCI) available * Patient must not have extensive intraabdominal adhesions that preclude safe placement of the intraperitoneal port