Docetaxel and Prednisone With or Without Vaccine Therapy in Treating Patients With Metastatic Hormone-Resistant Prostate Cancer

Inclusion Criteria: * Patient must have histologically confirmed diagnosis of prostate cancer (adenocarcinoma of the prostate) * Patient must have metastatic disease as evidenced by the presence of soft tissue and/or bone metastases on imaging studies (computed tomography \[CT\] of abdomen/pelvis, bone scintigraphy) * Patient must have castrate-resistant disease, defined as follows: * Patient must have received standard of care androgen deprivation treatment (ADT) before trial entry (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 * Patient must have been treated previously with a nonsteroidal antiandrogen, with evidence of subsequent disease progression; subjects must be off use of anti-androgen for at least 4 weeks (for flutamide) or 6 weeks (for bicalutamide or nilutamide) prior to randomization; 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 * Patient must have castration levels of testosterone (\< 50 ng/dL) within 4 weeks prior to randomization * Patient must have progressive disease while receiving ADT as defined by any one of the following as per the Prostate Cancer Clinical Trials Working Group 2 (PCWG2) criteria: * PSA: at least two consecutive rises in serum PSA, obtained at a minimum of 1-week intervals, and each 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 by RECIST criteria version 1.1; the greatest diameter of a target lesion must be at least 1.0 cm by CT scan (1.5 cm in shortest axis for lymph nodes) * Non-measurable (bone) disease: the appearance of two or more new areas of uptake on bone scan 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 \[MRI\]) * Patient must not have poor prognosis features suggested by the following required information: * Presence of visceral (non-lymph node, non-bone) metastases * Poor performance status (Eastern Cooperative Oncology Group \[ECOG\] performance status \[PS\] of 2 or greater) * Alkaline phosphatase (IU/L) \> 2 x institutional upper limit of normal * Lactate dehydrogenase (LDH) (U/L) \> 2 x institutional upper limit of normal * Patient must have an ECOG performance status of 0 or 1 * White blood cell (WBC) count \>= 2000/mm\^3 * Absolute neutrophil count (ANC) \>= 1500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Creatinine =\< 2.0 mg/dL * Aspartate aminotransferase (AST), alanine aminotransferase (ALT) =\< 1.5 x institutional upper limit of normal (ULN) * Total bilirubin \< institutional upper limit of normal (ULN) * Patient must have completed any prior treatments (apart from androgen deprivation as previously described) \>= 4 weeks prior to randomization and have recovered (to \< grade 2) from any acute toxicities attributed to this prior treatment * Patient must agree to use an accepted and effective method of contraception prior to study entry and for the duration of study participation (or at least 4 months after the last vaccination in subjects receiving vaccine series); if patient impregnates a woman while participating in this study, he should inform his treating physician immediately * Patient must not be receiving any other investigational agents or be receiving concurrent anticancer therapy other than ADT * Patient must not have been treated with a prior anti-cancer vaccine (including sipuleucel-T, Provenge®) * Patient must not have received treatment with any of the following medications within 4 weeks of randomization or while on study: * Systemic corticosteroids (excluding prednisone and dexamethasone administered as part of study protocol); inhaled, intranasal or topical corticosteroids are acceptable; steroid eye drops are contraindicated however, for 2 weeks prior to vaccination and for at least 4 weeks after vaccinia vaccination * PC-SPES * Saw palmetto * Megestrol * Ketoconazole * 5-alpha-reductase inhibitors - patients already taking 5-alpha-reductase inhibitors prior to 28 days prior to randomization may stay on these agents throughout the course of therapy, but these should not be started while patients are on study * Diethyl stilbestrol * Any other hormonal agent or supplement with possible anti-cancer activity * Patient must not have been treated with external beam radiation therapy within 4 weeks of randomization * Patient must not have received prior radiation therapy to \> 30% of bone marrow * Patient must not have had surgery within 4 weeks of randomization * Patient must not have received prior chemotherapy within 6 months of randomization; prior and/or concurrent treatment with bisphosphonates, however, is permitted * Patient must not have received prior chemotherapy for metastatic prostate cancer * Patient cannot have a known history of human immunodeficiency virus (HIV) 1 or 2, human T-lymphotropic virus (HTLV)-1, hepatitis B, or hepatitis C (or any other potentially immunosuppressive infection); eligible subjects must have negative serologic testing for HIV, hepatitis B surface antigen, and hepatitis C * Patient cannot have a history of autoimmune disease requiring active immunosuppressive therapy or have organ dysfunction \>= grade 2 as a result of known autoimmune disease; eligible subjects must have antinuclear antibodies (ANA) titer \< 1:320 * Patient must not have undergone splenectomy * Patient must not have other active malignancies other than non-melanoma skin cancers or carcinoma in situ of the bladder; subjects with a history of other cancers who have been adequately treated and have been recurrence-free for \>= 3 years are eligible * Patient cannot have a known allergy to eggs * Patient cannot have a known intolerance or allergic reactions to docetaxel or compounds of similar chemical or biologic composition * Patient cannot have a known history of allergy or intolerable reaction to a previous vaccinia virus vaccination (e.g., smallpox) * Patient or close household contacts of patient (those who share housing or have close physical contact with the patient) cannot have close physical contact to persons with the following conditions within 3 weeks after potential vaccinia immunization: * A history of eczema, active eczema or other acute, chronic or exfoliative skin conditions, including Darier's disease (e.g. atopic dermatitis, burns, impetigo, varicella zoster, severe acne, or open wounds) * Pregnant or nursing women * Children under 3 years of age * Immunodeficient or immunosuppressed persons (e.g. HIV, or treated for other diseases with immunosuppressive agents) * Any other moderate or severe acute illness until the illness resolves Patients who would be unable to avoid these conditions for a 3-week period are not eligible; patients should also refer to the patient instruction sheet for vaccinia virus * Patient cannot have known brain metastases * Patient cannot have a known history of recent (within 6 months) stroke, myocardial infarction, unstable angina, New York Heart Association class II-IV congestive heart failure, or significant cardiomyopathy requiring treatment * Patient cannot take known strong inducers or inhibitors of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) within 2 weeks of beginning docetaxel through its discontinuation; substrates of CYP3A4 with a narrow therapeutic/toxicity window may be used with caution, with prior approval of the study chair or institution's principal investigator (PI)