Vaccine Therapy and Pembrolizumab in Treating Patients With Hormone-Resistant, Metastatic Prostate Cancer

Inclusion Criteria:

* Histologically confirmed diagnosis of prostate cancer (adenocarcinoma of the prostate)
* Metastatic disease as evidenced by the presence of soft tissue and/or bone metastases on imaging studies (CT of abdomen/pelvis, bone scintigraphy)
* Castrate-resistant disease, defined as follows:

  * All patients must have received (and be receiving) standard of care androgen deprivation treatment (surgical castration versus gonadotropin-releasing hormone \[GnRH\] analogue or antagonist treatment); subjects receiving GnRH analogue or antagonist must continue this treatment throughout the time on this study
  * Patients may or may not have been treated previously with a nonsteroidal antiandrogen; for patients previously treated with an antiandrogen, they must be off use of anti-androgen for at least 4 weeks (for flutamide) or 6 weeks (for bicalutamide or nilutamide) prior to registration; moreover, subjects who demonstrate an anti-androgen withdrawal response, defined as a \>= 25% decline in PSA within 4-6 week of stopping a nonsteroidal antiandrogen, are not eligible until the PSA rises above the nadir observed after antiandrogen withdrawal
  * Patients must have a castrate serum level of testosterone (\< 50 ng/dL) within 6 weeks of day 1
* Progressive disease while receiving androgen deprivation therapy defined by any one of the following as per the Prostate Cancer Clinical Trials Working Group 2 (PCWG2) bone scan criteria or Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 during or after completing last therapy:

  * PSA: at least two consecutive rises in serum PSA, obtained at a minimum of 1-week intervals, with the final value \>= 2.0 ng/mL
  * Measurable disease: \>= 50% increase in the sum of the cross products of all measurable lesions or the development of new measurable lesions; the short axis of a target lymph node must be at least 15 mm by spiral CT to be considered a target lesion
  * Non-measurable (bone) disease: the appearance of two or more new areas of uptake on bone scan (or fluorine F 18 sodium fluoride \[NaF\] PET/CT) consistent with metastatic disease compared to previous imaging during castration therapy; the increased uptake of pre-existing lesions on bone scan will not be taken to constitute progression, and ambiguous results must be confirmed by other imaging modalities (e.g. X-ray, CT or magnetic resonance imaging \[MRI\])
* Prior treatment with abiraterone or enzalutamide is permitted, but patients must have been off prior corticosteroid treatment for at least 3 months
* Life expectancy of at least 6 months
* Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2
* White blood cells (WBC) \>= 2000/mm\^3
* Absolute neutrophil count (ANC) \>= 1000/mm\^3
* Hemoglobin (HgB) \>= 9.0 gm/dL
* Platelets \>= 100,000/mm\^3
* Creatinine =\< 2.0 mg/dL
* Aspartate aminotransferase (AST), alanine aminotransferase (ALT) =\< 2.5 x institutional upper limit of normal
* No known history of human immunodeficiency virus (HIV) 1 and 2, human T-lymphotropic virus (HTLV)-1, or active hepatitis B or hepatitis C
* Patients must be at least 4 weeks from any prior treatments and have recovered (to \< grade 2) from acute toxicity attributed to this prior treatment, unless considered chronic
* Patients must be willing and able (in the opinion of the treating physician) to undergo two research biopsies for the investigational component of this trial
* Patients must be willing to undergo two leukapheresis procedures for the investigational component of this trial
* Patients must be willing to undergo FLT PET/CT or NaF PET/CT scans for the investigational component of this trial and have no known allergies to FLT or NaF
* For those patients who are sexually active, they must be willing to use barrier contraceptive methods during the period of treatment on this trial (and for four weeks after the last DNA immunization treatment for patients in Arm 1)
* Patients must be informed of the experimental nature of the study and its potential risks, and must sign an Institutional Review Board (IRB)-approved written informed consent form indicating such an understanding

Exclusion Criteria:

* Small cell or other variant (non-adenocarcinoma) prostate cancer histology, unless there is evidence that the tumor expresses PAP
* Patients may not be receiving other investigational agents or be receiving concurrent anticancer therapy other than standard androgen deprivation therapy
* Concurrent bisphosphonate therapy is not excluded, however patients should not start bisphosphonate therapy while on this study; those patients already receiving bisphosphonate therapy should continue at the same dosing and schedule as prior to study entry
* Rapidly progressive symptomatic metastatic disease, as defined by the need for increased opioid analgesics within one month of registration for the treatment of pain attributed to a prostate cancer metastatic lesion; patients receiving opioids must receive approval from the principal investigator (PI) for eligibility
* Treatment with any of the following medications within 28 days of registration, or while on study, is prohibited:

  * Systemic corticosteroids (at doses over the equivalent of 5 mg prednisone daily) - not permitted within 3 months of registration; inhaled, intranasal or topical corticosteroids are acceptable
  * Prostate cancer (PC)-SPES
  * Saw palmetto
  * Megestrol
  * Ketoconazole
  * 5-alpha-reductase inhibitors - patients already taking 5-alpha-reductase inhibitors prior to 28 days prior to registration may stay on these agents throughout the course of therapy, but these should not be started while patients are on study
  * Diethyl stilbestrol
  * Abiraterone
  * Enzalutamide
  * Radium 223 (Xofigo)
  * Any other hormonal agent or supplement being used with the intent of cancer treatment
* External beam radiation therapy within 4 weeks of registration is prohibited, or anticipated need for radiation therapy (e.g. imminent pathological fracture or spinal cord compression) within 3 months of registration
* Major surgery within 4 weeks of registration is prohibited
* Prior cytotoxic chemotherapy (for example, but not limited to, docetaxel, mitoxantrone, cabazitaxel) within 6 months of registration is prohibited
* Patients with a history of life-threatening autoimmune disease
* Patients with a history of allergic reactions to filgrastim (GM-CSF) or the tetanus vaccine
* Patients who have undergone splenectomy
* Patients must not have other active malignancies other than non-melanoma skin cancers or superficial bladder cancer; subjects with a history of other cancers who have been adequately treated and have been recurrence-free for \>= 3 years are eligible
* Patients with known brain metastases
* Any antibiotic therapy or evidence of infection within 1 week of registration
* Any other medical intervention or condition, which, in the opinion of the PI or treating physician, could compromise patient safety or adherence with the study requirements (including biopsies or leukapheresis procedures) over the primary 3-6 month treatment period
* Patients cannot have concurrent enrollment on other phase I, II, or III investigational treatment studies

NOTE: There is no exclusion for prior immune-based therapy. This includes patients previously treated on Arms 1 or 2 who are otherwise eligible for treatment on Arm 3 or 4.