Tacrolimus and Mycophenolate Mofetil With or Without Sirolimus in Preventing Acute Graft-Versus-Host Disease in Patients Who Are Undergoing Donor Stem Cell Transplant for Hematologic Cancer

Inclusion Criteria: * Ages \> 50 years with hematologic malignancies treatable by unrelated hematopoietic cell transplant (HCT) * Ages =\< 50 years of age with hematologic diseases treatable by allogeneic HCT who through pre-existing medical conditions or prior therapy are considered to be at high risk for regimen related toxicity associated with a conventional transplant (\> 40% risk of transplant related mortality \[TRM\]) (This criterion can include patients with a HCT-comorbidity index (CI) score of \>= 1; transplants should be approved for these inclusion criteria by both the participating institutions' patient review committees such as the Patient Care Conference (PCC) at the Fred Hutchinson Cancer Research Center (FHCRC) and by the principal investigators at the collaborating centers) * Patients =\< 50 years of age who have received previous high-dose transplantation do not require patient review committee approvals (All children \< 12 years must be discussed with the FHCRC principal investigator (PI) \[Brenda Sandmaier, MD 206 6674961\] prior to registration) * Ages =\< 50 years of age with chronic lymphocytic leukemia (CLL); these patients do not require patient review committee approvals * Ages =\< 50 years of age with hematologic diseases treatable by allogeneic HCT who refuse a conventional HCT (Transplants must be approved for these inclusion criteria by both the participating institutions' patient review committee such as PCC at the FHCRC and by the principal investigators at the collaborating centers) * The following diseases will be permitted although other diagnoses can be considered if approved by PCC or the participating institutions' patient review committees and the principal investigators: * Aggressive non-Hodgkin lymphomas (NHL) and other histologies such as Diffuse large B cell NHL not eligible for autologous hematopoietic stem cell transplant (HSCT), not eligible for conventional myeloablative HSCT, or after failed autologous HSCT * Mantle Cell NHL may be treated in first complete response (CR) (Diagnostic lumbar puncture \[LP\] required pretransplant) * Low grade NHL with \< 6 month duration of CR between courses of conventional therapy * CLL must have either * Failed to meet National Cancer Institute (NCI) Working Group criteria for complete or partial response after therapy with a regimen containing fludarabine phosphate (FLU) (or another nucleoside analog, e.g. Cladribine \[2-CDA\], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing FLU (or another nucleoside analog); * Failed FLU-CY-Rituximab (FCR) combination chemotherapy at any time point; or * Have "17p deletion" cytogenetic abnormality; patients should have received induction chemotherapy but could be transplanted in 1st CR * Hodgkin Lymphoma must have received and failed frontline therapy * Multiple Myeloma must have received prior chemotherapy; consolidation of chemotherapy by autografting prior to nonmyeloablative HCT is permitted * Acute Myeloid Leukemia (AML) must have \< 5% marrow blasts at the time of transplant * Acute Lymphocytic Leukemia (ALL) must have \< 5% marrow blasts at the time of transplant * Chronic Myeloid Leukemia (CML) patients will be accepted if they are beyond chronic phase (CP)1 and if they have received previous myelosuppressive chemotherapy or HCT and have \< 5% marrow blasts at time of transplant * Myelodysplasia (MDS)/Myeloproliferative Syndrome (MPS) patients must have received previous myelosuppressive chemotherapy or HCT and have \< 5% marrow blasts at time of transplant * Waldenstrom's Macroglobulinemia must have failed 2 courses of therapy * DONOR: FHCRC matching allowed will be Grades 1.0 to 2.1: unrelated donors who are prospectively: * Matched for human leukocyte antigen (HLA)-A, B, C, DRB1 and DQB1 by high resolution typing * Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing * DONOR: Donors are excluded when preexisting immunoreactivity is identified that would jeopardize donor hematopoietic cell engraftment; this determination is based on the standard practice of the individual institution; the recommended procedure for patients with 10 of 10 HLA allele level (phenotypic) match is to obtain a panel reactive antibody (PRA) screens to class I and class II antigens for all patients before HCT; if the PRA shows \> 10% activity, then flow cytometric or B and T cell cytotoxic cross matches should be obtained; the donor should be excluded if any of the cytotoxic cross match assays are positive; for those patients with an HLA Class I allele mismatch, flow cytometric or B and T cell cytotoxic cross matches should be obtained regardless of the PRA results; a positive anti-donor cytotoxic crossmatch is an absolute donor exclusion * DONOR: Patient and donor pairs homozygous at a mismatched allele in the graft rejection vector are considered a two-allele mismatch, i.e., the patient is A\*0101 and the donor is A\*0102, and this type of mismatch is not allowed * DONOR: Only filgrastim (G-CSF) mobilized peripheral blood mononuclear cell (PBMC) only will be permitted as a HSC source on this protocol Exclusion Criteria: * Patients with rapidly progressive intermediate or high grade NHL * Patients with a diagnosis of chronic myelomonocytic leukemia (CMML) * Central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * Presence of circulating leukemic blasts (in the peripheral blood) detected by standard pathology for patients with AML, MDS, ALL or CML * Fertile men or women unwilling to use contraceptive techniques during and for 12 months following treatment * Females who are pregnant or breast-feeding * Patients with active non-hematological malignancies (except non-melanoma skin cancers) or those with non-hematological malignancies (except non-melanoma skin cancers) who have been rendered with no evidence of disease, but have a greater than 20% chance of having disease recurrence within 5 years * Fungal infections with radiological progression after receipt of amphotericin B or active triazole for greater than 1 month * Cardiac ejection fraction \< 35%; ejection fraction is required if age \> 50 years or there is a history of anthracycline exposure or history of cardiac disease * Diffusion capacity of carbon monoxide (DLCO) \< 40%, total lung capacity (TLC) \< 40%, forced expiratory volume in one second (FEV1) \< 40% and/or receiving supplementary continuous oxygen * The FHCRC PI of the study must approve of enrollment of all patients with pulmonary nodules * Patients with clinical or laboratory evidence of liver disease would be evaluated for the cause of liver disease, its clinical severity in terms of liver function, and the degree of portal hypertension; patients will be excluded if they are found to have fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bridging fibrosis, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3 mg/dL, or symptomatic biliary disease * Karnofsky score \< 60 or Lansky score \< 50 * Patient has poorly controlled hypertension and on multiple antihypertensives * Human immunodeficiency virus (HIV) positive patients * Active bacterial or fungal infections unresponsive to medical therapy * All patients receiving antifungal therapy voriconazole, posaconazole, or fluconazole and who are then randomized to ARM 3 must have rapamycin reduced according to the Standard Practice of Antifungal Therapy Guidelines * The addition of cytotoxic agents for cytoreduction with the exception of tyrosine kinase inhibitors (such as imatinib), cytokine therapy, hydroxyurea, low dose cytarabine, chlorambucil, or Rituxan will not be allowed within three weeks of the initiation of conditioning * DONOR: Donor (or centers) who will exclusively donate marrow * DONOR: Donors who are HIV-positive and/or, medical conditions that would result in increased risk for G-CSF mobilization and harvest of G-PBMC