This phase III trial compares immunotherapy drugs (nivolumab or brentuximab vedotin) when given with combination chemotherapy in treating patients with newly diagnosed stage III or IV classic Hodgkin lymphoma. Immunotherapy with monoclonal antibodies, such as nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Brentuximab vedotin is a monoclonal antibody, brentuximab, linked to a toxic agent called vedotin. Brentuximab attaches to cancer cells in a targeted way and delivers vedotin to kill them. Chemotherapy drugs, such as doxorubicin, vinblastine, and dacarbazine, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. The addition of nivolumab or brentuximab vedotin to combination chemotherapy may shrink the cancer or extend the time without disease symptoms coming back.