This partially randomized phase II trial studies how well brentuximab vedotin or crizotinib and combination chemotherapy works in treating patients with newly diagnosed stage II-IV anaplastic large cell lymphoma. Monoclonal antibody-drug conjugates, such as brentuximab vedotin, can block cancer growth in different ways by targeting certain cells. Crizotinib and methotrexate may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy 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. It is not yet known whether brentuximab vedotin and combination chemotherapy is more effective than crizotinib and combination chemotherapy in treating anaplastic large cell lymphoma.