This phase II/III trial compares the effect of adding chemotherapy before and after surgery versus after surgery alone (usual treatment) in treating patients with stage II-III gallbladder cancer. Chemotherapy drugs, such as gemcitabine and cisplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving chemotherapy before surgery may make the tumor smaller; therefore, may reduce the extent of surgery. Additionally, it may make it easier for the surgeon to distinguish between normal and cancerous tissue. Giving chemotherapy after surgery may kill any remaining tumor cells. This study will determine whether giving chemotherapy before surgery increases the length of time before the cancer may return and whether it will increase a patients life span compared to the usual approach.

This phase II/III compares the standard therapy (chemotherapy plus cetuximab) versus adding bevacizumab to standard chemotherapy, versus combination of just bevacizumab and atezolizumab in treating patients with head and neck cancer that has spread to other places in the body (metastatic or advanced stage) or has come back after prior treatment (recurrent). Immunotherapy with monoclonal antibodies, such as atezolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Bevacizumab is in a class of medications called antiangiogenic agents. It works by stopping the formation of blood vessels that bring oxygen and nutrients to tumor. This may slow the growth and spread of tumor. Cetuximab is in a class of medications called monoclonal antibodies. It binds to a protein called EGFR, which is found on some types of cancer cells. This may help keep cancer cells from growing. Cisplatin and carboplatin are in a class of chemotherapy medications known as platinum-containing compounds. They work by killing, stopping, or slowing the growth of cancer cells. Docetaxel is in a class of chemotherapy medications called taxanes. It stops cancer cells from growing and dividing and may kill them. The addition of bevacizumab to standard chemotherapy or combination therapy with bevacizumab and atezolizumab may be better than standard chemotherapy plus cetuximab in treating patients with recurrent/metastatic head and neck cancers.

This phase II/III trial studies how well sentinel lymph node biopsy works and compares sentinel lymph node biopsy surgery to standard neck dissection as part of the treatment for early-stage oral cavity cancer. Sentinel lymph node biopsy surgery is a procedure that removes a smaller number of lymph nodes from your neck because it uses an imaging agent to see which lymph nodes are most likely to have cancer. Standard neck dissection, such as elective neck dissection, removes many of the lymph nodes in your neck. Using sentinel lymph node biopsy surgery may work better in treating patients with early-stage oral cavity cancer compared to standard elective neck dissection.