VEGF Trap and Docetaxel in Treating Patients With Persistent or Recurrent Ovarian Epithelial Cancer, Primary Peritoneal Cancer, or Fallopian Tube Cancer

Criteria: * Histologically confirmed ovarian epithelial, primary peritoneal, or fallopian tube cancer: Persistent or recurrent disease * Measurable disease, defined as \>= 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques: * Must have \>= 1 target lesion to assess response; * Tumors within a previously irradiated field are considered nontarget lesions * Must have received 1 prior platinum-based chemotherapy regimen (for primary disease) containing carboplatin, cisplatin, or other organoplatinum compound: Initial treatment may have included any of the following: * High-dose therapy; * Consolidation therapy; * Extended therapy administered after surgical or nonsurgical assessment * AND Must have received 1 prior platinum-based chemotherapy regimen (for primary disease) containing carboplatin, cisplatin, or other organoplatinum compound: One additional cytotoxic regimen for recurrent or persistent disease allowed * No history or evidence of CNS disease, including primary brain tumor or brain metastases * Zubrod performance status 0-2 (0-1 for patients who received 2 prior regimens \[taxane and/or platinum regimens are counted separately\]) * Absolute neutrophil count \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Hemoglobin \>= 9 g/dL * Bilirubin normal * aspartate aminotransferase / alanine aminotransferase (AST/ALT) \< 2.5 times upper limit of normal (ULN) * Creatinine \< 1.5 times ULN OR creatinine clearance \> 60 mL/min * Prothrombin Time (PT)/international normalized ratio (INR) \< 1.5 OR in-range INR 2-3 (if patient is on a stable dose of therapeutic warfarin or low molecular weight heparin) * PTT \< 1.2 times control * Urine protein:creatinine ratio \< 1 * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception during and for at least 6 months after completion of study treatment * No active infection requiring antibiotics * No sensory and motor neuropathy \>= grade 2 * No history of allergic reactions attributed to compounds of similar chemical or biologic composition to VEGF Trap, Magnevist, or fludeoxyglucose F 18 * No history of allergic reaction to paclitaxel or docetaxel or to products mixed in Cremophor EL or Tween 80 * No active bleeding or pathologic condition that would carry a high risk of bleeding, including any of the following: Known bleeding disorder; Coagulopathy; Peptic ulcer disease; Diverticulitis; Tumor involving major vessels * No active and/or untreated pulmonary embolism, deep vein thrombosis, or other thromboembolic event (i.e., any condition associated with aberrant clotting or migration of an induced clot) * No history or evidence of other CNS disease, including any of the following: Seizures not controlled with standard medical therapy; Cerebrovascular accident; Transient ischemic attack; Subarachnoid hemorrhage within the past 6 months * No clinically significant cardiovascular disease, including any of the following: Uncontrolled hypertension (i.e., systolic blood pressure \[BP\] \> 150 mm Hg or diastolic BP \> 100 mm Hg; systolic BP \> 180 mm Hg and diastolic BP \< 90 mm Hg OR diastolic BP \> 90 mm Hg on \>= 2 measurements within the past 3 months); Myocardial infarction; Coronary or peripheral artery bypass graft * No clinically significant cardiovascular disease, including any of the following: New York Heart association class III or IV congestive heart failure; Serious cardiac arrhythmia requiring medication; Peripheral vascular disease \>= grade 2; Unstable angina within the past 6 months; Clinically significant peripheral artery disease (e.g., claudication) within the past 6 months * No known hypersensitivity to Chinese hamster ovary cell products or other recombinant human or humanized antibodies * Able to undergo an MRI scan: No claustrophobia; No implanted devices or metallic foreign bodies not compatible with MRI (e.g., ferromagnetic implants or pacemakers); No known history of allergic reaction to gadolinium contrast agents * No other invasive malignancy within the past 5 years except nonmelanoma skin cancer * No significant traumatic injury within the past 28 days * Recovered from prior therapy to NCI Common Terminology Criteria for Adverse Events (CTCAE) version 3.0 grade =\< 1: Alopecia allowed * No prior VEGF Trap * No prior cancer treatment that would preclude study treatment * Prior paclitaxel allowed * Prior docetaxel for primary or recurrent disease allowed provided the following criteria are met: No disease progression during therapy; No disease relapse within 3 months of completing therapy; No persistent disease at the completion of primary therapy * More than 7 days since prior placement of a vascular access device or core biopsy * At least 1 week since prior hormonal therapy for the malignant tumor * At least 4 weeks since other prior therapy, including immunologic agents (6 weeks for nitrosoureas or mitomycin C) * More than 28 days since prior major surgery, open biopsy, dental extraction, or other dental surgery/procedure resulting in an open wound * No concurrent major surgery * No concurrent combination antiretroviral therapy for HIV-positive patients * No other concurrent anticancer therapy * No other concurrent investigational agents * No concurrent hematopoietic growth factors during course 1 of phase I * Concurrent hormone replacement therapy allowed * White blood count (WBC) \>= 3,000/mm\^3