Sorafenib Tosylate, Cisplatin, and Docetaxel in Treating Patients With Recurrent or Metastatic Squamous Cell Carcinoma of the Head and Neck

Inclusion Criteria: * Patients must have histologic or cytologic proof of recurrent/metastatic squamous cell carcinoma of the head and neck (SCCHN) of any primary site, including unknown primary cancers of the head and neck excluding nasopharyngeal carcinoma of histologic types World Health Organization (WHO) 2 or 3, paranasal sinuses primary or squamous cell carcinoma that originated in the skin * Patients must have SCCHN that is either (a) recurrent, judged incurable by surgery or (chemo)radiation or (b) metastatic * Patients must not have received prior chemotherapy for recurrent or metastatic disease * Patients may have received one regimen of induction, concomitant chemoradiotherapy and/or adjuvant chemotherapy as part of their initial treatment with curative intent, which must have been completed for a minimum of 4 months prior to study treatment and patient must have been progression-free for at least 4 months since completion of treatment with curative intent * Patients with recurrent disease are allowed a maximum of one prior radiation therapy regimen, either curative or palliative * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0-1 * Patients must have measurable disease using Response Evaluation Criteria in Solid Tumors (RECIST) criteria; patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Absolute neutrophil count (ANC) 1500/mm\^3 * Platelet count 100,000/mm\^3 * Creatinine within normal limits (WNL), or creatinine clearance \>= 60 ml/min, based on the Cockroft-Gault formula * Total bilirubin WNL * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) less than twice the upper normal limit * Patients must have controlled blood pressure (150/90) prior to initiation of treatment * Women must not be pregnant or breast feeding; women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of sorafenib administration * Patients must be human immunodeficiency virus (HIV)-negative * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 3 months prior to entering the study * Patients who are receiving any other investigational agents * Patients with known brain metastases will be excluded from this clinical trial * History of allergic reactions attributed to compounds of similar chemical or biologic composition to sorafenib, docetaxel, cisplatin * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Uncontrolled hypertension (systolic pressure \> 140 mm Hg or diastolic pressure \> 90 mm Hg \[National Cancer Institute (NCI)-Common Terminology Criteria for Adverse Events (CTCAE) version (v)4.0\] on repeated measurement) despite optimal medical management * Evidence or history of bleeding diathesis or coagulopathy * Subject with any pulmonary hemorrhage/bleeding event of NCI-CTCAE v4.0 grade 2 or higher within 4 weeks before randomization; any other hemorrhage/bleeding event of NCI-CTCAE v4.0 grade 3 or higher within 4 weeks before randomization * Subjects with thrombotic, embolic, venous, or arterial events, such as cerebrovascular accident (including transient ischemic attacks) within 6 months of informed consent * Subjects who have used strong cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) inducers (e.g., phenytoin, carbamazepine, phenobarbital, St. John's Wort \[Hypericum perforatum\], or rifampin \[rifampicin\], and/or rifabutin) within 28 days before randomization * Subjects with any previously untreated or concurrent cancer that is distinct in primary site or histology from breast cancer except cervical cancer in-situ, treated basal cell carcinoma, or superficial bladder tumor; subjects surviving a cancer that was curatively treated and without evidence of disease for more than 3 years before randomization are allowed; all cancer treatments must be completed at least 3 years prior to study entry (i.e., signature date of the informed consent form) * History of organ allograft; (including corneal transplant) * Any malabsorption condition * Inability to comply with the protocol * Any condition which, in the investigator's opinion, makes the subject unsuitable for trial participation