Paclitaxel With or Without Viral Therapy in Treating Patients With Recurrent or Persistent Ovarian Epithelial, Fallopian Tube, or Primary Peritoneal Cancer

Inclusion Criteria: * Patients must have recurrent or persistent epithelial ovarian, fallopian tube or primary peritoneal carcinoma; histologic documentation of the original primary tumor is required via the pathology report * Patients must have measurable disease or detectable (non-measurable) disease: * Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded); each lesion must be \>= 10 mm when measured by computed tomography \[CT\], magnetic resonance imaging \[MRI\] or caliper measurement by clinical exam; or \>= 20 mm when measured by chest x-ray; lymph nodes must be \> 15 mm in short axis when measured by CT or MRI * Detectable (non-measurable) disease is defined as not having measurable disease but has at least one of the following conditions: * Baseline values of CA-125 at least 2 x upper limit of normal (ULN); * Ascites and/or pleural effusion attributed to tumor; * Solid and/or cystic abnormalities on radiographic imaging that do not meet RECIST 1.1 definitions for target lesions * Patient with measurable disease must have at least one "target lesion" to be used to assess response on this protocol as defined by RECIST 1.1; tumors within a previously irradiated field will be designated as "non-target" lesions unless progression is documented or a biopsy is obtained to confirm persistence at least 90 days following completion of radiation therapy * Patients must not be eligible for a higher priority Gynecologic Oncology Group (GOG) protocol, if one exists; in general, this would refer to any active GOG phase III protocol or rare tumor protocol for the same patient population * Patients who have received one prior regimen must have a GOG performance status of 0, 1, or 2 * Patients who have received two or three prior regimens must have a GOG performance status of 0 or 1 * Recovery from effects of recent surgery, radiotherapy, or chemotherapy: * Patients should be free of active infection requiring antibiotics (with the exception of uncomplicated urinary tract infection \[UTI\]) * Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration; continuation of hormone replacement therapy is permitted * Any other prior therapy directed at the malignant tumor, including chemotherapy, biologic/targeted and immunologic agents, must be discontinued at least three weeks prior to registration * Patients must have had one prior platinum-based chemotherapeutic regimen for management of primary disease containing carboplatin, cisplatin, or another organoplatinum compound; this initial treatment may have included intraperitoneal therapy, consolidation, non-cytotoxic agents (biologic/targeted) or extended therapy administered after surgical or non-surgical assessment; if patients were treated with paclitaxel for their primary disease, this can have been given weekly or every 3 weeks * Patients are allowed to receive, but are not required to receive, two additional cytotoxic regimens for management of recurrent or persistent disease, with no more than 1 non-platinum, non-taxane regimen; treatment with weekly paclitaxel for recurrent or persistent disease is NOT allowed * Patients are allowed to receive, but are not required to receive, non-cytotoxic (biologic/targeted) therapy as part of their primary treatment regimen; patients are allowed to receive, but are not required to receive, non-cytotoxic (biologic/targeted) therapy as part of their treatment for recurrent or persistent disease and/or as treatment for recurrent or persistent disease; if non-cytotoxic (biologic/targeted) therapy is given alone (i.e., not in combination with cytotoxic chemotherapy) it will NOT count as a prior regimen * For the purposes of this study, poly (adenosine diphosphate \[ADP\]-ribose) polymerase (PARP) inhibitors will NOT count as a prior regimen when given alone (i.e., not in combination with cytotoxic chemotherapy) * Patients who have received only one prior cytotoxic regimen (platinum-based regimen for management of primary disease), must have a platinum-free interval of less than 12 months, or have progressed during platinum-based therapy, or have persistent disease after a platinum-based therapy * Absolute neutrophil count (ANC) greater than or equal to 1,500/mcl * Platelets greater than or equal to 100,000/mcl * Hemoglobin greater than or equal to 9 g/dL * Creatinine less than or equal to 1.5 x institutional upper limit normal (ULN) * Bilirubin less than or equal to 1.5 x ULN * Serum glutamic oxaloacetic transaminase (SGOT) less than or equal to 3 x ULN * Alkaline phosphatase less than or equal to 2.5 x ULN * Neuropathy (sensory and motor) less than or equal to grade 1 * Patients of childbearing potential must have a negative pregnancy test prior to the study entry and be practicing an effective form of contraception; (pregnant women are excluded from this study) * Patients must have signed an approved informed consent and authorization permitting the release of personal health information * Patients must meet pre-entry requirements as specified * Patients must be able to avoid direct contact with severely immune-compromised individuals such as patients who have had a recent bone-marrow or organ transplant or patients with acquire immunodeficiency syndrome (AIDS); contact should be avoided on the days of Reolysin treatment and for the 2 days following Reolysin treatment * Patients must be able to avoid direct contact with pregnant or nursing women and infants while receiving Reolysin; contact should be avoided on the days of Reolysin treatment and for the 2 days following Reolysin treatment Exclusion Criteria: * Patient who have had previous treatment with Reolysin or other oncolytic virus; patients who have had previous treatment with weekly paclitaxel for recurrent or persistent disease * Patients with a history of other invasive malignancies, with the exception of non-melanoma skin cancer and other specific malignancies are excluded if there is any evidence of other malignancy being present within the last three years; patients are also excluded if their previous cancer treatment contraindicates this protocol therapy * Patients who have received prior radiotherapy to any portion of the abdominal cavity or pelvis OTHER THAN for the treatment of ovarian, fallopian tube, or primary peritoneal cancer within the last three years are excluded; prior radiation for localized cancer of the breast, head and neck, or skin is permitted, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease * Patients who have received prior chemotherapy for any abdominal or pelvic tumor OTHER THAN for the treatment of ovarian, fallopian tube, or primary peritoneal cancer within the last three years are excluded; patients may have received prior adjuvant chemotherapy for localized breast cancer, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease * Patients with a past history of primary endometrial cancer are excluded unless all of the following conditions are met: stage not greater than I-B; no more than superficial myometrial invasion, without vascular or lymphatic invasion; no poorly differentiated subtypes, including papillary serious, clear cell or other International Federation of Gynecology and Obstetrics (FIGO) grade 3 lesions * Patients with known human immunodeficiency virus (HIV) or hepatitis B or C are excluded due to risk of viral infectivity of Reolysin therefore patients with a pre-existent infection are not eligible * Patients who are receiving immunosuppressive therapy including chronic oral steroids (at an equivalent dose of greater than prednisone 5 mg daily) * Women who are pregnant or nursing; pregnant women are excluded from this study; breastfeeding should be discontinued while the mother is being treated with the agents in this clinical trial * Myocardial infarction or unstable angina within 6 months of the first date of study therapy * History of serious ventricular arrhythmia (i.e., ventricular tachycardia or ventricular fibrillation) or cardiac arrhythmias requiring anti-arrhythmic medications (except for atrial fibrillation that is well controlled with antiarrhythmic medication) * Troponin \> ULN * Baseline ejection fraction \< 50% as assessed by echocardiogram or multi gated acquisition scan (MUGA) * New York Heart Association (NYHA) class II or greater congestive heart failure * History of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within six months of the first date of study therapy