Chemotherapy With or Without Bevacizumab in Treating Patients With Recurrent or Metastatic Head and Neck Squamous Cell Carcinoma

Inclusion Criteria: * Patients must have histologically or cytologically confirmed squamous cell cancer of the head and neck (SCCHN), from any primary site, including unknown primary cancers of the head and neck; patient must not have nasopharyngeal carcinoma of histologic types World Health Organization (WHO) 2 or 3 or squamous cell carcinoma that originated in the skin * Patients must have SCCHN that is either (a) recurrent, judged incurable by surgery or radiation or (b) metastatic; NOTE: Patients who refuse radical resection for recurrent disease are eligible; NOTE: A second primary squamous cell carcinoma of the head and neck is allowed if eligibility is based on a recurrent or metastatic first primary squamous cell carcinoma of the head and neck * No prior chemotherapy or biologic/molecular targeted therapy for recurrent or metastatic SCCHN * Patients may have received one regimen of induction, concomitant chemoradiotherapy and/or adjuvant chemotherapy as part of initial potential curative therapy but must not have received prior chemotherapy for recurrent or metastatic disease * A minimum of 4 months is required between last dose of chemotherapy or chemoradiotherapy and study treatment; in addition patients must be progression-free for at least 4 months after completion of chemotherapy or chemoradiotherapy or radiation plus cetuximab given with a curative intent; (cetuximab therapy: 4 months is required between last dose of chemotherapy or chemoradiotherapy and study treatment if part of concurrent regimen, 8 weeks if part of adjuvant regimen post radiation) * Patients having progression after 2 cycles of induction chemotherapy are not eligible for the study * No prior bevacizumab is allowed * A maximum of one prior radiotherapy regimen, curative or palliative, to the head and neck is allowed; if the radiation is combined with chemotherapy and/or cetuximab, a minimum of 4 months must elapse between the end of radiotherapy and registration; if the radiation is given alone, a minimum of 8 weeks must elapse between the end of radiotherapy and registration; a minimum of 3 weeks must elapse between prior radiation to other areas and registration * Patients must not be receiving any other investigational agent while on the study * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Patients must have recovered to grade 1 or better from any acute effects of prior surgery, chemotherapy, or radiation therapy, and should be \> 4 weeks post surgery; chronic late xerostomia, speech and swallowing abnormalities resulting from prior radiation or surgery are permitted if nutritional status is stable * Patients must have measurable disease based on Response Evaluation Criteria in Solid Tumors (RECIST); baseline measurements and evaluations of all sites of disease must be obtained =\< 4 weeks prior to randomization; disease in previously irradiated sites is considered measurable if there has been unequivocal disease progression or biopsy-proven residual carcinoma following radiation therapy; persistent disease after radiotherapy must be biopsy proven at least 8 weeks after completion of radiation therapy; (radiographic findings are acceptable providing that clear-cut measurements can be made) * Absolute neutrophil count (ANC) \>= 1500/mm\^3 * Hemoglobin (Hgb) \>= 8.0 g/dL * Platelet count \>= 100,000/mm\^3 * Creatinine clearance of \>= 60 ml/min; creatinine clearance may be measured or calculated; if calculating, creatinine clearance, use the Cockroft-Gault formula * Total bilirubin within normal limits (must be obtained =\< 2 weeks prior to randomization) * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) must be within the range allowing for eligibility * Alkaline phosphatase normal AND AST or ALT =\< 5 x upper limit of normal (ULN) * Alkaline phosphatase \> 1 but =\< 2.5 x ULN AND AST or ALT \> 1 but =\< 1.5 x ULN * Alkaline phosphatase \> 2.5 but =\< 5 x ULN AND AST or ALT normal * Alkaline phosphatase must be within the range allowing for eligibility * Urine dipstick must be =\< 0-1+ within 2 weeks (14 days) of randomization; if urine dipstick result is \> 1+, a calculation of urine protein creatinine (UPC) ratio is required; patients must have a UPC ratio \< 1.0 to participate in the study; NOTE: UPC ratio of spot urine is an estimation of the 24 urine protein excretion - a UPC ratio of 1 is roughly equivalent to a 24-hour urine protein of 1 gm * No known brain metastases * Patients who meet the following criteria will be excluded: * Tumors that invade major vessels (e.g. the carotid) as shown unequivocally by imaging studies * Central (i.e. within 2 cm from the hilum) lung metastases that are cavitary as shown unequivocally by imaging studies * Any prior history of bleeding related to the current head and neck cancer * History of gross hemoptysis (bright red blood of 1/2 teaspoon or more per episode of coughing) =\< 3 months prior to enrollment * No history of coagulopathy or hemorrhagic disorders * Patients should not have a history of thrombosis (e.g. pulmonary embolism or deep venous thrombosis) currently requiring therapeutic anticoagulation (prophylactic use of warfarin 1 mg per day is allowed) and international normalized ratio (INR) should be \< 1.5 at registration * Patients must not be receiving chronic daily treatment with aspirin (\> 325 mg/day) or non-steroidal anti-inflammatory agents (NSAID's) known to inhibit platelet function; the use of anti-platelet agents (e.g. dipyridamole \[Persantine\], ticlopidine \[Ticlid\], clopidogrel \[Plavix\]) is allowed only if patient is not receiving aspirin or NSAID's known to inhibit platelet function * No hypercalcemia related to head and neck cancer * Patients with a prior history of squamous cell or basal carcinoma of the skin or in situ cervical cancer must have been curatively treated; patients with a history of other prior malignancy must have been treated with curative intent and must have remained disease-free for 3 years post diagnosis * No current peripheral neuropathy \>= grade 2 at time of randomization * Patients must not have any co-existing condition that would preclude full compliance with the study * No prior history of severe hypersensitivity reaction to docetaxel or other drugs formulated with polysorbate 80, if the physician's choice of chemotherapy regimen is docetaxel * All patients must have blood pressure =\< 150/90 =\< 2 weeks prior to randomization; patients with history of hypertension must be well-controlled upon study entry (=\< 150/90) on a stable regimen of anti-hypertensive therapy * No major surgical procedure, open biopsy, or significant traumatic injury within 28 days prior to study enrollment, or anticipation of need for major surgical procedure during the course of the study * No unstable angina or myocardial infarction within the previous 6 months; no symptomatic congestive heart failure, New York Heart Association (NYHA) grade II or greater; no history of aortic dissection or presence of aneurysm \> 6 cm (or at high risk for rupture); no serious cardiac arrhythmia requiring medication (history of chronic atrial fibrillation or other atrial arrhythmia with controlled rate on medication is allowed); no clinically significant peripheral vascular disease manifested by intermittent claudication or need for vascular intervention; no history of aortic dissection; no history of any central nervous system (CNS) cerebrovascular ischemia or stroke within the last 6 months; no active serious infection * Patients should not have prior history of a serious human anti-human antibody (HAHA) reaction; patients with known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies are not eligible * Women must not be pregnant or breast feeding; pregnant women are excluded from this study; women of child-bearing potential and men must agree to total abstinence or to use adequate hormonal or barrier method of birth control prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while in this study, she should inform her treating physician immediately; all females of childbearing potential must have a blood test or urine study within 2 weeks prior to randomization to rule out pregnancy * Human immunodeficiency virus (HIV)-positive patients receiving combination anti-retroviral therapy are excluded from the study; appropriate studies will be undertaken in patients receiving combination anti-retroviral therapy when indicated * No history of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 6 months prior to registration