This phase III trial studies whether inotuzumab ozogamicin added to post-induction chemotherapy for patients with High-Risk B-cell Acute Lymphoblastic Leukemia (B-ALL) improves outcomes. This trial also studies the outcomes of patients with mixed phenotype acute leukemia (MPAL), and B-lymphoblastic lymphoma (B-LLy) when treated with ALL therapy without inotuzumab ozogamicin. Inotuzumab ozogamicin is a monoclonal antibody, called inotuzumab, linked to a type of chemotherapy called calicheamicin. Inotuzumab attaches to cancer cells in a targeted way and delivers calicheamicin to kill them. Other drugs used in the chemotherapy regimen, such as cyclophosphamide, cytarabine, dexamethasone, doxorubicin, daunorubicin, methotrexate, leucovorin, mercaptopurine, prednisone, thioguanine, vincristine, and pegaspargase or calaspargase pegol 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. This trial will also study the outcomes of patients with mixed phenotype acute leukemia (MPAL) and disseminated B lymphoblastic lymphoma (B-LLy) when treated with high-risk ALL chemotherapy. The overall goal of this study is to understand if adding inotuzumab ozogamicin to standard of care chemotherapy maintains or improves outcomes in High Risk B-cell Acute Lymphoblastic Leukemia (HR B-ALL). The first part of the study includes the first two phases of therapy: Induction and Consolidation. This part will collect information on the leukemia, as well as the effects of the initial treatment, in order to classify patients into post-consolidation treatment groups. On the second part of this study, patients will receive the remainder of the chemotherapy cycles (interim maintenance I, delayed intensification, interim maintenance II, maintenance), with some patients randomized to receive inotuzumab. Other aims of this study include investigating whether treating both males and females with the same duration of chemotherapy maintains outcomes for males who have previously been treated for an additional year compared to girls, as well as to evaluate the best ways to help patients adhere to oral chemotherapy regimens. Finally, this study will be the first to track the outcomes of subjects with disseminated B-cell Lymphoblastic Leukemia (B-LLy) or Mixed Phenotype Acute Leukemia (MPAL) when treated with B-ALL chemotherapy.