Nivolumab With or Without Ipilimumab in Treating Younger Patients With Recurrent or Refractory Solid Tumors or Sarcomas

Inclusion Criteria: * Parts A \& C: patients must be \>= 12 months and \< 18 years of age at the time of study enrollment * Parts B1-B6, B8, D1-D6, E3, E4: patients must be \>= 12 months and =\< 30 years of age at the time of study enrollment * Part B7: patients must be \>= 12 months and \< 18 years of age at the time of study enrollment * Patients must have had histologic verification of malignancy at original diagnosis or relapse * Parts A \& C: patients with recurrent or refractory solid tumors, without central nervous system (CNS) tumors or known CNS metastases, are eligible; note: CNS imaging for patients without a known history of CNS disease is only required if clinically indicated * Part B1: patients with relapsed or refractory neuroblastoma * Part B2: patients with relapsed or refractory osteosarcoma * Part B3: patients with relapsed or refractory rhabdomyosarcoma * Part B4: patients with relapsed or refractory Ewing sarcoma or peripheral primitive neuroectodermal tumor (PNET) * Part B5: patients with relapsed or refractory Hodgkin lymphoma * Part B6: patients with relapsed or refractory non-Hodgkin lymphoma * Part B7: patients with unresectable melanoma or metastatic melanoma or relapsed melanoma or refractory melanoma * Part B8: Patients with relapsed or refractory neuroblastoma (MIBG evaluable disease without Response Evaluation Criteria in Solid Tumors \[RECIST\] measurable lesion) * Once the dose-escalation portion of Part A is completed, c…