Study of Novel Therapies for Young People With Recurrent/Progressive Atypical Teratoid Rhabdoid Tumor (ATRT)

Inclusion Criteria: * In addition to the below, investigators are to refer to arm-specific inclusion criteria in the appendix. * Participants must have a pathologic diagnosis of central nervous system (CNS) ATRT, with confirmation of SWI/SNF-related, matrix-associated, actin-dependent regulator of chromatin, subfamily b, member 1 (SMARCB1) (INI1) loss by immunohistochemistry (IHC) and/or biallelic loss of function of SMARCB1 by molecular report. Loss of SWI/SNF Related, Matrix Associated, Actin Dependent Regulator of Chromatin, Subfamily A, Member 4 (SMARCA4) as confirmed by IHC or molecular report is also acceptable but requires study chair approval. * Participants must have confirmation of methylation report, co-enrollment on PNOC-030 or sufficient tumor tissue available for methylation-based subgrouping * Participants must have recurrent or progressive ATRT. * Participants age must be ≥1 and ≤ 39 years at the time of study enrollment. Please refer to arm specific inclusion criteria for potential variations in lower age eligibility limit. * Prior Therapy: Participants must have fully recovered from the acute effects of prior anti-cancer therapy, and the following wash-out periods need to be observed prior to enrollment: * Systemic myelosuppressive therapy: ≥ 21 days after the last dose (42 days for nitrosoureas or mitomycin C). * Intrathecal/intraventricular chemotherapy: \> 7 days after the last dose. * Small molecule/targeted/biologic agent: ≥ 7 days after the last dose. * Monoclonal antibodies: ≥ 21 days after the last dose. Other non-myelosuppressive anti-cancer agents: ≥ 3 drug half-lives after the last dose. * CAR-T cell therapy (systemic or intraventricular): \> 21 days. • Previous radiotherapy. Participants will be eligible following radiotherapy, if they meet the following criteria: * Previous craniospinal or total body radiotherapy: Participants must have received their last fraction ≥ 12 weeks prior to enrollment and have evidence of progressive/recurrent evaluable disease post radiation. * Previous focal radiotherapy to target lesions: Participants must have received their last fraction to target lesions ≥12 weeks prior to enrollment and have evidence of progressive/recurrent evaluable disease post radiation; investigators are reminded to review potentially eligible cases to avoid confusion with pseudo-progression. * Focal radiotherapy to non-target lesions: Participants may have received radiotherapy to nontarget lesions as long as the last fraction was \> 14 days prior to enrollment. Participants must have at least one non-irradiated lesion that is evaluable for response. * Performance Score: Karnofsky ≥ 50 for participants \> 16 years of age and Lansky ≥ 50 for participants ≤ 16 years of age. Participants who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. * Corticosteroids: Participants who are receiving corticosteroids must be on a stable or decreasing dose for at least 1 week prior to enrollment. Please also refer to arm-specific inclusion criteria for potential variations in steroid limitations. * Organ Function Requirements. Adequate Bone Marrow Function Defined as: * Peripheral absolute neutrophil count (ANC) ≥ 750/mm3 * Platelet count ≥ 75,000/cubic millimeters (mm3) (transfusion independent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment). Adequate Renal Function Defined as: * Serum creatinine ≤ 1.5 Upper Limit Normal (ULN) based on age and gender Adequate Liver Function Defined as: * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) for age; in presence of Gilbert's syndrome, total bilirubin \< 3 x ULN or direct bilirubin \< 1.5 x ULN * Alanine aminotransferase (ALT) ≤ 3 x ULN * Aspartate aminotransferase (AST) ≤ 3 x ULN Adequate Neurologic Function Defined as: * Participants with seizure disorder may be enrolled if well controlled. See arm-specific recommendations for potential interactions between anticonvulsant agent(s) with study drug. * Effect on the developing human fetus Recommendations on the potential effect of interventional agents on the developing human fetus will be specified in each study arm's details of therapeutic agents. Unless otherwise specified, the effects of study interventions should be considered potentially teratogenic. Thus, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study treatment and four months after its completion. Should a woman become pregnant or suspect she is pregnant while she is participating in this study, or should a male participant's partners become pregnant during study participation, they should inform the treating physician immediately. * A legal parent/guardian or patient must be able to understand, and willing to sign, a written informed consent and assent document, as appropriate. * Participants must enroll on Pediatric Neuro-oncology Consortium (PNOC) COMP if PNOC COMP is open to accrual at the enrolling institution. Exclusion Criteria: * Evidence of synchronous tumors or other extra-CNS malignancy * Participants who are receiving any other investigational agents * Participants who are currently receiving other anti-cancer agents * Participants with uncontrolled infection or other uncontrolled systemic illness * Female participants of childbearing potential who are pregnant or breast-feeding. Female participants of childbearing potential must have a negative serum or urine pregnancy test prior to the start of therapy and throughout study treatment. * History of allergic reactions attributed to compounds of similar chemical or biologic composition as the intended treatment regimen, as detailed in that arm's treatment description. Arm A Inclusion Criteria: * Subjects must meet all inclusion criteria for the overall study. * Patients must be evaluable per Response Assessment in Pediatric Neuro-Oncology (RAPNO) criteria for medulloblastoma and other leptomeningeal seeding tumors to be evaluated for the primary endpoint (Warren et al. 2018); patients with evaluable but non-measurable disease, including leptomeningeal disease or positive CSF cytology only are eligible. Patients with recurrent or progressive ATRT who receive surgery only for their disease progression and do not have evaluable disease may be eligible for study treatment but would not be included towards the primary efficacy endpoint (to be discussed with study chairs). * Subjects must be able to swallow intact capsules. * Adequate Metabolic Function Defined as: * Non-fasting glucose ≤ 140 milligrams per deciliter (mg/dL) without the use of antihyperglycemic agents. * If non-fasting glucose \> 140 mg/dL, a fasting glucose should be done. If fasting glucose ≤ 125 mg/dL without the use of antihyperglycemic agents, participant will meet adequate metabolic function criteria. * Triglycerides of \< 300 mg/dl and total cholesterol of \< 300 mg/dl - can be on lipid lowering medications as needed to achieve. * Adequate Cardiac Function Defined as: * Electrocardiogram (ECG) must be obtained to verify the Corrected QT Interval (QTC). If an abnormal reading is obtained, the ECG should be repeated in triplicate. * QTC \< 470 millisecond (msec) Arm A Exclusion Criteria: In addition to not meeting any of the exclusion criteria of the overall study, participation on arm A will also require that subjects do not meet any of the following: * Previous exposure to gemcitabine or paxalisib * Concomitant use of an antihyperglycemic agent (e.g. metformin) * Chronic diarrhea greater than Grade 2