Testing a New Combination of Anti-cancer Drugs in Patients Newly Diagnosed With Ewing Sarcoma Who Have Cancer That Has Spread to Other Parts of the Body

Inclusion Criteria: * All patients must be enrolled on APEC14B1 and consented to the Molecular Characterization Initiative (Part A) prior to enrollment and treatment on AEWS2431 * Patients must be ≥ 12 months to ≤ 50 years of age at time of enrollment * Newly diagnosed Ewing sarcoma and other round cell sarcomas as follows. For the purposes of eligibility, the following pathology diagnoses are eligible and molecular confirmation is not required to enroll: * Histologically confirmed Ewing sarcoma * Suspected Ewing sarcoma with molecular confirmation pending * Suspected high grade round cell sarcomas/sarcomas with eligible molecular alterations, pending molecular confirmation * Round cell sarcoma consistent with Ewing sarcoma * Round cell sarcoma not otherwise specified * Round cell sarcoma * Round cell sarcomas with EWSR1-non-ETS fusion * CIC-rearranged sarcoma * Sarcoma with BCOR genetic alterations * Patients with the following pathologic diagnoses that are known to contain EWSR1 or FUS fusions are not eligible: * Angiomatoid fibrous histiocytoma * Extraskeletal myxoid chondrosarcoma * Desmoplastic small round cell tumor * Clear cell sarcoma * Myxoid liposarcoma * If clinical molecular testing that reports both fusion partners has been successfully completed by the site prior to enrollment, patients may only enroll if that testing reported one of the eligible fusions * All patients must have evidence of distant metastatic disease. Biopsy of metastatic sites is not required. For this study, distant metastatic disease is defined as one or more of the following: * Lesions which are discontinuous from the primary tumor, are not regional lymph nodes, and do not share a bone or body cavity with the primary tumor. Skip lesions in the same bone as the primary tumor do not constitute metastatic disease. Skip lesions in an adjacent bone are considered bone metastasis. If there is any doubt whether lesions are metastatic, a biopsy of those lesions should be performed * Contralateral pleural effusion and/or contralateral pleural nodules * Distant lymph node involvement * Patients with pulmonary nodules are considered to have metastatic disease if the patient has: * Solitary nodule ≥ 0.5 cm or multiple nodules of ≥ 0.3 cm unless lesion is biopsied and negative for tumor * Patients with solitary nodule \< 0.5 cm or multiple nodules \< 0.3 cm are not considered to have lung metastasis unless biopsy documents tumor * Bone marrow metastatic disease (bone marrow disease) is based on morphologic evidence of Ewing sarcoma based on hematoxylin and eosin (H\&E) stains. In the absence of morphologic evidence of marrow involvement on H\&E, patients with bone marrow involvement detected ONLY by flow cytometry, reverse transcriptase-polymerase chain reaction (RT-PCR), fluorescence in situ hybridization (FISH), immunohistochemistry or PET-CT will NOT be considered to have clinical bone marrow involvement for the purposes of this study * Creatinine clearance or radioisotope glomerular filtration rate (GFR) ≥ 50 mL/min/1.73 m\^2 or serum creatinine based on age/sex as follows: * Age: 1 to \< 2 years: Male 0.6; female 0.6 * Age: 2 to \< 6 years: Male 0.8; female 0.8 * Age: 6 to \< 10 years: Male 1; female 1 * Age: 10 to \< 13 years: Male 1.2; female 1.2 * Age: 13 to \< 16 years: Male 1.5; female 1.4 * Age: ≥ 16 years: Male 1.7; female 1.4 OR * A 24-hour urine creatinine clearance ≥ 50 mL/min/1.73 m\^2 OR * A GFR ≥ 50 mL/min/1.73 m\^2. GFR must be performed using direct measurement with a nuclear blood sampling method OR direct small molecule clearance method (iothalamate or other molecule per institutional standard) * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) for age * Serum glutamic pyruvic transaminase (SGPT) (alanine aminotransferase \[ALT\]) ≤ 3 x ULN for age (except for patients with liver metastasis who may enroll if ALT ≤ 5 times ULN for age) * Note: For the purpose of this study, the ULN for SGPT (ALT) has been set to the value of 45 U/L * Shortening fraction of ≥ 27% by echocardiogram, or ejection fraction of ≥ 50% by echocardiogram * No known congenital QT syndrome * No known family history of congenital QT syndrome * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better * Known HIV-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial Exclusion Criteria: * Patients with regional node involvement as their only site of disease beyond the primary tumor * Patients whose primary tumors arise in the intra-dural soft tissue (e.g., brain and spinal cord) * Note: metastatic disease is allowable * Patients with known Charcot-Marie-Tooth disease * Patients who have had complete or partial resection of the primary tumor at initial diagnosis will only be eligible if adequate imaging (CT or MRI for most primary tumor sites) was obtained prior to surgery * Patients who have received prior chemotherapy for current diagnosis, except for patients who have started cycle 1 VDC post-consent and within the timelines allowed for * Patients who have received prior radiation therapy for current diagnosis * Patients previously treated with a multitargeted tyrosine kinase inhibitor * History of organ allograft (including allogeneic bone marrow transplant) * Known hypersensitivity to regorafenib * Active or chronic hepatitis B or C, or chronic hepatitis B or C requiring treatment with antiviral therapy * Patients receiving strong CYP3A4 inducers or strong CYP3A4 inhibitors * Female patients who are pregnant since fetal toxicities and teratogenic effects have been noted for several of the study drugs. A pregnancy test is required for female patients of childbearing potential * Lactating females who plan to breastfeed their infants * Females of childbearing potential must agree to either practice medically accepted highly-effective methods of contraception or abstain from heterosexual intercourse for the duration of the protocol therapy through 12 months after the last dose of cyclophosphamide or ifosfamide, 6 months after the last dose of doxorubicin, etoposide, and irinotecan, and 7 months after the last dose of regorafenib, whichever is longer * Male patients with female partners of childbearing potential must agree to either practice a medically accepted highly-effective methods of contraception or abstain from heterosexual intercourse for the duration of the protocol therapy through 6 months after the last dose of doxorubicin and ifosfamide, 4 months after the last dose of cyclophosphamide, etoposide and regorafenib, and 3 months after the last dose of irinotecan, whichever is longer * All patients and/or their parents or legal guardians must sign a written informed consent * All institutional, Food and Drug Administration (FDA), and National Cancer Institute (NCI) requirements for human studies must be met