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