Low-Dose Selinexor and Choline Salicylate for Non-Hodgkin or Hodgkin Lymphoma, Histiocytic/Dendritic Cell Neoplasm, or Relapsed or Refractory Multiple Myeloma

Inclusion Criteria:

* Age \>= 18 years
* Non Hodgkin or Hodgkin lymphoma or histiocytic/dendritic cell neoplasm meeting one of the following criteria:

  * Biopsy-proven relapsed and/or refractory Non-Hodgkin or Hodgkin lymphoma or histiocytic/dendritic cell neoplasms

    * Relapsed is defined as a relapse that occurred after having a response to the last therapy that lasted \> 26 weeks
    * Refractory is no response (stable disease or progressive disease while on therapy) or relapse within 6 months. Refractoriness to autologous stem cell transplant will be defined as disease progression within 52 weeks following transplant OR

Multiple myeloma neoplasm meeting the following criteria:

* Relapsed and/or refractory multiple myeloma (RRMM) as per the International Myeloma Working Group (IMWG) uniform criteria
* If extramedullary myeloma, most recent tumor biopsy must be \< 26 weeks prior to registration

  * Measurable or assessable disease:
* For Non-Hodgkin or Hodgkin Lymphoma and histiocytic/dendritic cell:

  * Measurable disease is defined as measurable by computed tomography (CT) \[dedicated CT or the CT portion of a positron emission tomography (PET)/CT\] or MRI: To be considered measurable, there must be at least one lesion that has a single diameter of \>= 1.5 cm NOTE: Skin lesions can be used if the area is \>= 1.5 cm in at least one diameter and photographed with a ruler. Patients with assessable disease by PET/CT are also eligible as long as the assessable disease is biopsy proven lymphoma or histiocytic/dendritic cell neoplasms
* For Multiple myeloma:
* Measurable disease by IMWG criteria as defined by at least one of the following:

  * Serum M-protein \>= 0.5 g/dL
  * Urine M-protein \>= 200 mg in a 24-hour collection
  * Serum Free Light Chain level \>= 10 mg/dL provided the free light chain ratio is abnormal
  * Measurable plasmacytoma (at least one lesion that has a single diameter of \>= 2 cm on CT portion of PET/CT scan or MRI)
  * Bone marrow plasma cells \>= 10%

    * Exception: A patient with non-secretory multiple myeloma (MM) but with bone marrow plasma cells \>= 30% may be considered for enrollment after discussion with the PI that includes the feasibility of an individualized plan for response assessment
  * Patients with Immunoglobulin A (IgA) or Immunoglobulin D (IgD) myeloma in whom serum protein electrophoresis is deemed unreliable, due to co-migration of normal serum proteins with the para protein in the beta region, may be considered eligible as long as total serum IgA or IgD level is elevated above normal range

    * Patients with non-Hodgkin or Hodgkin lymphoma must have previously been treated with \>= 2 lines of therapy
    * Patients with histiocytic/dendritic cell neoplasms must previously have been treated with \>= 1 line of therapy
    * Patients with RRMM must have received ≥4 prior therapies whose disease is refractory to \>= 2 proteasome inhibitors, \>= 2 immunomodulatory agents, and an anti-CD38 monoclonal antibody
    * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0, 1, or 2
    * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 (obtained =\< 14 days prior to registration)
    * Platelet count \>= 100,000/mm\^3 (obtained =\< 14 days prior to registration)
    * Hemoglobin \>= 8.5 g/dL (may be transfused to reach criteria) (obtained =\< 14 days prior to registration)
    * Total bilirubin \< 2 x upper limit of normal (ULN) (or total bilirubin =\< 3.0 x ULN with direct bilirubin =\< 1.5 x ULN in patients with well-documented Gilbert's syndrome) (obtained =\< 14 days prior to registration)
    * Aspartate transaminase (AST) =\< 2.5 x ULN and alanine aminotransferase (ALT) =\< 2.5 x ULN (obtained =\< 14 days prior to registration)
    * Calculated creatinine clearance must be \>= 35 ml/min using the Cockcroft Gault formula (obtained =\< 14 days prior to registration)
    * Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only
    * Female of childbearing potential (FCBP\*) must commit to take highly effective contraceptive precautions\*\* without interruption during the study and continue for at least 12 weeks after the last dose of selinexor and CS. FCBP must refrain from breastfeeding and donating oocytes during the course of the study. Males must use an effective barrier method of contraception without interruption during the study and continue for at least 12 weeks after the last dose of selinexor and CS. They must refrain from donating sperm during the study participation.
* \*FCBP defined as sexually mature women who have not undergone bilateral tubal ligation, bilateral oophorectomy, or hysterectomy; or who have not been postmenopausal (i.e., who have not menstruated at all) for at least 1 year

  * Highly effective forms of birth control are methods that achieve a failure rate of less than 1% per year when used consistently and correctly. Highly effective forms of birth control include: hormonal contraceptives (oral, injectable, patch, and intrauterine devices), male partner sterilization, or total abstinence from heterosexual intercourse, when this is the preferred and usual lifestyle of the patient NOTE: The double-barrier method (e.g., synthetic condoms, diaphragm, or cervical cap with spermicidal foam, cream, or gel), periodic abstinence (such as calendar, symptothermal, post-ovulation), withdrawal (coitus interruptus), lactational amenorrhea method, and spermicide-only are not acceptable as highly effective methods of contraception

    * Provide written informed consent
    * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study)
    * Willingness to provide mandatory blood specimens per protocol for Pharmacokinetics (PKs) and banking
    * For lymphoma, extramedullary myeloma and histiocytic/dendritic cell neoplasms, willing to provide mandatory tissue samples for correlative research. NOTE: If an institution is not able to provide the tissue, it does not cause the patient to be ineligible; however, the collection of these tissues is strongly recommended.

Exclusion Criteria:

* Any of the following because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown:

  * Pregnant women
  * Nursing women
* Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens
* Patients known to have active hepatitis B, or C infection, or known to be positive for hepatitis C virus (HCV) ribonucleic acid (RNA) or hepatitis B surface antigen (HBsAg) \[hepatitis B virus (HBV) surface antigen\]. Patients known to be human immunodeficiency virus (HIV) positive, except those with CD4+ T-cell (CD4+) counts \>= 350 cells/microliter (µL) and on an established antiretroviral therapy (ART) for at least twelve weeks and have an HIV viral load less than 400 copies/mL prior to enrollment
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Life expectancy of \< 6 months
* Active gastrointestinal (GI) dysfunction interfering with the ability to swallow tablets, or any GI dysfunction that could interfere with absorption of study treatment
* Known intolerance to or contraindications for choline salicylate therapy. Patients with known allergy to acetylsalicylic acid (ASA) are not eligible
* Prior exposure to a selective inhibitors of nuclear export (SINE) compound, including selinexor
* Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm
* Active second malignancy requiring treatment that would interfere with the assessment of the response of the primary cancer to this protocol therapy. Patients with treated malignancies on hormonal therapy (for example breast or prostate cancer) are eligible
* History of myocardial infarction =\< 6 months, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias
* Radiation, chemotherapy, or immunotherapy or any other anticancer therapy =\< 2 weeks prior to registration. NOTE: Exception: patients on any Bruton tyrosine kinase (BTK) inhibitor (ibrutinib, zanabrutinib, acalabrutinib, etc), or venetoclax, or corticosteroids (any dose) may continue therapy up until the new regimen has started at investigator discretion. After the start of protocol therapy, corticosteroids can be used at investigator's discretion and tapered to lowest possible dose
* Active graft versus (vs.) host disease (after allogeneic stem cell transplantation) at registration
* Major surgery (including bowel resection) =\< 3 weeks prior to registration
* Must not be currently eligible or have declined high-dose therapy with autologous stem cell transplantation rescue or chimeric antigen receptor (CAR)-T cell therapy
* Primary mediastinal (thymic) large B-cell lymphoma (PMBL)
* Known active central nervous system (CNS) lymphoma. Patients with previous CNS involvement can enroll if the CNS component is inactive
* Patients who are on active anticoagulant therapy with direct oral anticoagulants (DOACs), aspirin or warfarin are not eligible due to potential bleeding. EXCEPTIONS: Patients who are on aspirin (81 mg) for primary prevention of cardiovascular disease can enroll, but the ASA needs to be held while on this protocol therapy