Pomalidomide and Dose-Adjusted EPOCH +/- Rituximab for HIV-Associated Lymphomas

* INCLUSION CRITERIA: * Histologically or cytologically confirmed B-cell NHL confirmed by the Laboratory of Pathology (LP), NCI, with one or more of the following features: * Leptomeningeal/CSF involvement * High-risk for CNS relapse per CNS-IPI (score 4-6) * Plasmablastic histology * Gamma herpesvirus positive tumor * Presence of KS * Measurable or evaluable lymphoma. * Positive HIV1/2 serology. * Individuals may not have received prior curative-intent chemotherapy for lymphoma. Individuals who have received prior treatment as a bridge to curative-intent therapy will be considered per Protocol Chair discretion if \>= 2 weeks since administration. Steroids given for any reason or rituximab given for multicentric Castleman disease may be given any time prior to treatment start. * Age \>=18 years * Eastern Cooperative Oncology Group performance status (ECOG-PS) \<=4 * Individuals of childbearing potential (IOCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 25 mIU/mL within 2 weeks prior to and again within 1 day before starting the study drugs and must either commit to continued abstinence from penetrative vaginal intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before the participant starts taking pomalidomide and for 12 months after the last dose of combined chemotherapy. * Individuals able to father a child must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and up to six (6) months after the last dose of the study drug(s). We also will recommend individuals able to father a child with IOCBP partners to ask the partners to be on an effective birth control (hormonal, intrauterine device (IUD), surgical sterilization). Individuals able to father a child must not freeze or donate sperm within the same period. * All individuals must agree to be registered into the mandatory POMALYST REMS(R)TM program and be willing and able to comply with the requirements of the POMALYST REMS(R)TM program. * Able to take aspirin 81mg orally daily or another substitute thromboprophylaxis. * Adequate organ and marrow function as defined below unless abnormalities are attributed to lymphoma or HIV as determined by investigator: * absolute neutrophil count \>=1,000/mcL * platelets \>=75,000/mcL * total bilirubin \<=1.5 X institutional upper limit of normal (individuals with history of Gilbert disease are eligible if total bilirubin \<= 5 mg/dL with \<80% unconjugated bilirubin) * aspartate aminotransferase (AST) / alanine transaminase (ALT) \<=3 X institutional upper limit of normal * creatinine clearance \>=60 mL/min/1.73 m\^2 for individuals with creatinine levels above institutional normal. * Hepatitis B virus (HBV) infection must be on suppressive antiviral therapy. * Willingness to take and adhere to ART (individuals are not required to be on any specific regimen of ART). * Individuals must understand and sign a written informed consent document. EXCLUSION CRITERIA: * Individuals may not receive investigational agents on other clinical trials. * Requirement of any of the agents listed as prohibited thearapies. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to pomalidomide or other agents used in study. * Parenchymal brain involvement with lymphoma. * Ejection fraction less than 40% by echocardiography (ECHO) * CTCAEv5.0 Grade 3-4 neuropathy * History of malignant tumors other than KS or KSHV-associated multicentric Castleman Disease, (MCD), unless: * In complete remission for \>= 1 year from the time response was first documented; or, * Completely resected basal cell carcinoma; or, * In situ squamous cell carcinoma of the cervix or anus; or, * Prior or concurrent malignancy has a natural history or treatment which does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen per Protocol Chair discretion. * Known drug-related, inherited, or acquired procoagulant disorder including prothrombin gene mutation 20210, antithrombin III deficiency, protein C deficiency, protein S deficiency and antiphospholipid syndrome but not including heterozygosity for the Factor V Leiden mutation or the presence of a lupus anticoagulant in the absence of other criteria for the antiphospholipid syndrome. * Symptomatic congestive heart failure * Unstable angina pectoris, symptomatic cardiac arrhythmia, or cardiac arrhythmia requiring medical treatment. * Uncontrolled intercurrent illness or participants considered to be of poor medical health due to a serious, uncontrolled medical disorder, non-malignant systemic disease or active uncontrolled infection (excluding lymphoma or HIV) as documented in prior records or suggested by medical history, physical examination or standard clinical assessments such as imaging and laboratory studies. * Pregnant or nursing individuals (if lactating, must agree not to nurse while taking pomalidomide).