Radiation Therapy, Paclitaxel, and Carboplatin With or Without Trastuzumab in Treating Patients With Esophageal Cancer

Inclusion Criteria: * Pathologically confirmed primary adenocarcinoma of the esophagus that involves the mid (up to 25 cm), distal, or esophagogastric junction; the cancer may involve the stomach up to 5 cm * Endoscopy with biopsy * PRIOR TO STEP 1 REGISTRATION BUT WITHIN 56 DAYS PRIOR TO STEP 2 REGISTRATION * Intent to submit tissue for central HER2 testing * Stage T1N1-2, T2-3N0-2, according to the American Joint Committee on Cancer (AJCC) 7th edition staging, based on the following minimum diagnostic work-up: * Chest/abdominal/pelvic computed tomography (CT) or whole-body positron emission tomography (PET)/CT (NOTE: if CT is performed at this time point, whole-body PET/CT will be required prior to step 2 registration; PET/CT of skull base to mid-thigh is acceptable) (NOTE: if adenopathy is noted on CT or whole-body PET/CT scan, an endoscopic ultrasound is not required prior to STEP 2 registration as long as adequate tissue has been obtained for central HER2 testing) * Patients may have regional adenopathy including para-esophageal, gastric, gastrohepatic and celiac nodes; if celiac adenopathy is present, it must be =\< 2 cm * Patients with tumors at the level of the carina or above must undergo bronchoscopy to exclude fistula * Zubrod performance status 0-2 * Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 * Platelets \>= 100,000 cells/mm\^3 * Hemoglobin \>= 8.0 g/dL (note: the use of transfusion or other intervention to achieve hemoglobin \[Hgb\] \>= 8.0 g/d…