Bortezomib, Rituximab, and Dexamethasone With or Without Temsirolimus in Treating Patients With Untreated or Relapsed Waldenstrom Macroglobulinemia or Relapsed or Refractory Mantle Cell or Follicular Lymphoma

Inclusion Criteria:

* Histologically proven diagnosis
* For phase I portion (Arm A, B, C and D), patients must have of one of the following:

  * Relapsed Waldenstrom's macroglobulinemia
  * Relapsed/refractory mantle cell lymphoma; previous treatment with at least one standard regimen and no longer responsive to that regimen
  * Relapsed/refractory follicular lymphoma; previous treatment with at least one standard regimen and no longer responsive to that regimen
  * Relapsed/refractory marginal zone lymphoma; previous treatment with at least one standard regimen and no longer responsive to that regimen
  * Relapsed/refractory small lymphocytic lymphoma; previous treatment with at least one standard regimen and no longer responsive to that regimen
* For phase II portion (Arm E and F), patients must have a diagnosis of symptomatic Waldenstrom's macroglobulinemia, either untreated or relapsed, confirmed by the presence of all of the following:

  * Bone marrow lymphoplasmacytosis with

    * \>= 10% lymphoplasmatic cells (measured within 28 days prior to registration) OR
    * Aggregates or sheets of one of the following: lymphocytes, plasma cells or lymphoplasmacytic cells on the bone marrow biopsy (measured within 28 days prior to registration)
  * Measurable disease defined as a quantitative immunoglobulin M (IgM) monoclonal protein of \>= 1000 mg/dL obtained within 28 days prior to registration
  * Cluster of differentiation 20 (CD20) positive bone marrow or lymph node by immunohistochemistry or flow cytometry obtained within 28 days prior to registration
  * Lymph node biopsy must be done =\< 28 days prior to registration if used as an eligibility criterion for study entry
  * Serum protein electrophoresis (SPEP) is required to be performed within 28 days prior to registration
* Additional requirements for Waldenstrom's macroglobulinemia (WM) patients (phase I and II):
* In addition to measurable disease, patients must have symptomatic disease defined by one or more of the following:

  * Laboratory studies defining eligibility (hemoglobin \[Hgb\], platelet count, viscosity) must have been obtained within 28 days prior to registration; if more than one test was obtained, the most recent one will be utilized
  * Hemoglobin =\< 11 g/dL
  * Hyperviscosity syndrome or measured viscosity level of \>= 4 centipoise

    * NOTE: for these patients it is strongly recommended that they undergo therapeutic plasmapheresis prior to initiation of therapy
  * Platelet count \< 100,000/mm\^3
  * Symptomatic lymphadenopathy, splenomegaly, or hepatomegaly
  * Constitutional symptoms including fever, night sweats, or unexplained weight loss (at least 10% of body weight in \< 6 months)
  * Symptomatic cryoglobulinemia
* Additional requirements for WM patients (phase I):

  * Patients must have received previous treatment with at least one standard regimen and are no longer responsive to that regimen
  * There must have been at least 21 days since the last regimen and patient must have recovered from any previous treatment-related toxicity to =\< grade 1
* Additional requirements for WM patients (phase II):

  * For previously treated patients, no more than 4 prior regimens are allowed
  * If last regimen is with rituximab there must have been at least 6 months since last rituximab dose, and if without rituximab there must have been at least 3 months since last regimen
* For all phase I patients, there must have been at least 21 days since last regimen and any previous non-hematologic treatment related toxicity must have resolved to =\< grade 1
* Patients must not be receiving concurrent steroids \> 10 mg prednisone (or equivalent) per day
* Prior irradiation is allowed if \>= 28 days prior to registration have elapsed since the date of last treatment
* Fasting serum cholesterol =\< 300 mg/dL OR =\< 7.75 mmol/L AND fasting triglycerides =\< 2.5 x institutional upper limit of normal (ULN), within 28 days prior to registration

  * NOTE: in case one or both of these thresholds are exceeded, the patient can only be included after initiation of appropriate lipid lowering medication; patients cannot be enrolled if they do not meet these criteria on or off lipid lowering medication; patients must start lipid lowering medication and cholesterol and triglycerides must be below said levels before study entry
* Patients must not have had prior exposure to mammalian target of rapamycin (m-TOR) inhibitors (sirolimus, temsirolimus, everolimus)
* Women must not be pregnant or breast-feeding; all females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy; a female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months)
* Women of childbearing potential and sexually active males must use an accepted and effective method of contraception throughout the study and for 8 weeks following discontinuation of everolimus
* Patients must have no history of prior malignancy except for adequately treated basal cell or squamous cell skin cancer or in-situ cervical cancer; the patient may also have had other cancer for which the patient was curatively treated with surgery alone and from which the patient has been disease free for \>= 5 years
* Platelets \>= 75,000 mm\^3
* Neutrophils \>= 1,000 mm\^3
* Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) and serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 2.5 x institutional ULN
* Direct bilirubin =\< 1.5 mg/dL
* Serum creatinine =\< 2.5 mg/dL
* Patients must be tested for hepatitis B surface antigen (HBsAg) and hepatitis B core antibody (anti-HBc) within 28 days of registration and will not be eligible if found to be positive
* Patients must not have any severe and/or uncontrolled medical condition or other conditions that could affect their participation in the study, including, but not restricted to:

  * Symptomatic congestive heart failure of New York Heart Association class III or IV
  * Unstable angina pectoris, symptomatic congestive heart failure, myocardial infarction within 3 months of start of study treatment, serious uncontrolled cardiac arrhythmia or any other clinically significant heart disease
  * Severely impaired lung function as defined as spirometry and diffusing capacity of the lung for carbon monoxide (DLCO) (corrected for Hgb) that is \< 50% of the normal predicted value and/or oxygen (O2) saturation \< 88% at rest on room air
  * Active (acute or chronic) or uncontrolled severe infections
* Patients must have Eastern Cooperative Oncology Group (ECOG)-American College of Radiology Imaging Network (ACRIN) performance status of =\< 2
* Patients must not have grade 2 or higher neuropathy
* Patients must not have concurrent use of angiotensin-converting enzyme (ACE) inhibitors (angioedema), and no concurrent use of strong cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducers and inhibitors