Sorafenib Tosylate, Combination Chemotherapy, Radiation Therapy, and Surgery in Treating Patients With High-Risk Stage IIB-IV Soft Tissue Sarcoma

Inclusion Criteria: * Histologically confirmed, soft-tissue sarcoma: excluding rhabdomyosarcoma (pleomorphic rhabdomyosarcoma patients are eligible), Ewing's, primitive neuroectodermal tumor (PNET), osteosarcoma, or gastrointestinal stromal tumor * American Joint Committee on Cancer (AJCC) (7th edition) stage IIb, III, or IV patients planned for resection of the primary tumor * \> 5 cm in greatest dimension * Intermediate or high-grade * Superficial or deep * Sarcoma located on upper (includes shoulder) or lower (includes hip) extremities or on body wall * Intermediate or high-grade: grades 2 or 3 on scale of 1-3 * Left ventricular ejection fraction (LVEF) \>= 50% * Absolute neutrophil count (ANC) \>= 1500/uL * Hemoglobin (Hgb) \>= 9.0 g/dL * Platelets \>= 100,000/uL * Creatinine =\< 1.5 x upper limit of normal (ULN) * Bilirubin =\< 1.5 mg/dL * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 1.5 x ULN * International normalized ratio (INR) \< 1.5 or a prothrombin time (PT)/partial thromboplastin time (PTT) within normal limits; patients receiving anti-coagulation treatment with an agent such as warfarin or heparin may be allowed to participate; for patients on warfarin, the INR should be measured prior to initiation of sorafenib and monitored at least weekly, or as defined by the local standard of care, until INR is stable * No prior chemotherapy, radiation, or biotherapy * No major surgery within 4 weeks prior to study entry * No contraindications to limb-sparing surgery; patient should be evaluated by a surgeon who specializes in sarcoma resections prior to study enrollment to ensure patient (pt) is a candidate for limb-sparing surgery * No severe peripheral vascular disease * Adequate contraception must be used and patients must not be pregnant or breastfeeding; women of childbearing potential and men must agree to use adequate contraception (barrier method of birth control) prior to study entry and for the duration of study participation; men and women should use adequate birth control for at least thirty days after the last administration of sorafenib * Women of childbearing potential must have negative serum pregnancy test performed within 7-days prior to registration * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1 * Patient must sign a study-specific consent form prior to registration * Ability to understand and the willingness to sign a written informed consent; a signed informed consent must be obtained prior to any study specific procedures Exclusion Criteria: * Patients with known brain metastases; patients with neurological symptoms must undergo a computed tomography (CT) scan/magnetic resonance imaging (MRI) of the brain to exclude brain metastases * Uncontrolled intercurrent illness including, but not limited to, ongoing or active serious infection \> Common Terminology Criteria for Adverse Events (CTCAE) grade 2, symptomatic congestive heart failure, unstable angina pectoris, cardiac ventricular arrhythmia requiring anti-arrhythmic therapy, or psychiatric illness/social situations that would limit compliance with study requirements; patients must not have unstable angina (angina symptoms at rest) or new onset angina (began within the last 3 months) or myocardial infarction within the past 6 months * Uncontrolled hypertension defined as systolic blood pressure \> 150 mmHg or diastolic pressure \> 90 mmHg, despite optimal medical management * Known human immunodeficiency virus (HIV) infection or chronic hepatitis B or C * Thrombolic or embolic events such as a cerebrovascular accident including transient ischemic attacks within the past 6 months * Pulmonary hemorrhage/bleeding event \>= CTCAE grade 2 within 4 weeks of first dose of study drug * Any other hemorrhage/bleeding event \>= CTCAE grade 3 within 4 weeks of first dose of study drug * Serious non-healing wound, ulcer, or bone fracture * Evidence or history of bleeding diathesis or coagulopathy * Major surgery or significant traumatic injury within 4 weeks of first study drug * Use of strong cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducers * Known or suspected allergy to sorafenib or any agent given in the course of this trial * Any condition that impairs patient's ability to swallow whole pills * Any malabsorption problem * Pregnant or lactating women are excluded from this study * Patients with a "currently active" second malignancy other than non-melanoma skin cancers are not to be registered; patients are not considered to have a "currently active" malignancy if they have completed therapy and considered by their physician to be at less than 30% risk of relapse * Any uncontrolled thyroid disease * Requirement for hemodialysis or peritoneal dialysis