Sorafenib and Temsirolimus in Treating Patients With Metastatic, Recurrent, or Unresectable Melanoma

Inclusion Criteria: * Histologically or cytologically confirmed melanoma, meeting 1 of the following criteria: recurrent or unresectable stage III disease, stage IV disease, non-choroidal origin. * Tumor must be accessible to biopsy unless appropriate tumor sample collection has occurred within the past 3 months and patient agrees to provide these samples for this study. * Eastern Cooperative Oncology Group (ECOG) performance status 0-1. * Bilirubin normal * Creatinine normal or creatinine clearance ≥ 60 mL/min * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception during and for 30 days after completion of study treatment. * No history of allergic reactions attributed to compounds of similar chemical or biological composition to sorafenib or temsirolimus. * No uncontrolled hypertension, defined as systolic blood pressure \> 140 mm Hg on 2 separate days \< 1 week prior to study entry OR diastolic pressure \> 90 mm Hg on 2 separate days \< 1 week prior to study entry. * No evidence of bleeding diathesis or coagulopathy. * No condition that would impair the ability to swallow pills (e.g., gastrointestinal tract disease resulting in an inability to take oral medication; requirement for IV alimentation; or active peptic ulcer disease). * No uncontrolled illness including, but not limited to, any of the following: ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia or psychiatric illness or social situations that would limit study compliance. * No traumatic injury within the past 3 weeks. * No more than 1 prior systemic chemotherapy regimen for metastatic melanoma (Phase II). * No prior sorafenib, temsirolimus, or any other agents targeting raf, vascular endothelial growth factor (VEGF)/VEGF receptor, or mTOR (Phase II). * No prior surgical procedures affecting absorption. * At least 3 weeks since prior major surgery. * At least 4 weeks since prior chemotherapy (6 weeks for nitrosoureas or mitomycin C) for melanoma and recovered. * At least 4 weeks since prior radiotherapy and recovered. * Prior biologic or immunotherapeutic regimens allowed. * Prior regional chemotherapy regimens (e.g., isolated limb perfusion) allowed but only 1 prior regional chemotherapy regimen allowed if all target lesions are within the prior regional treatment field. * No concurrent enzyme-inducing antiepileptic drugs including, but not limited to, any of the following: phenytoin, carbamazepine, phenobarbital, rifampin or Hypericum perforatum (St. John's wort). * No concurrent prophylactic hematopoietic colony-stimulating factors. * No other concurrent investigational agents. * No other concurrent anticancer agents or therapies for this cancer. * No concurrent full-dose anticoagulation (i.e., warfarin, IV heparin, or low-molecular weight heparin). * No concurrent grapefruit or grapefruit juice. * No concurrent combination antiretroviral therapy for HIV-positive patients. * Concurrent prophylactic anticoagulation therapy (e.g., low-dose warfarin) allowed provided prothrombin time (PT) international normalized ratio (INR) \< 1.1 times upper limits of normal (ULN). * Unidimensionally measurable disease \>= 20 mm by conventional techniques or \>= 10 mm by spiral computed tomography (CT) scan (longest diameter to be recorded) and margins of visible cutaneous metastatic lesions should be clearly defined and measured in at least one dimension as \>= 10 mm. * No known brain metastases unless the following criteria are met: no radiographical evidence of recurrences in the brain \>= 3 months after complete resection of the brain metastases, asymptomatic brain metastases stable for \>= 3 months since whole-brain radiation therapy and/or stereotactic radiosurgery and must not require steroid for brain metastases. * White Blood Count (WBC) \>= 3,000/mm³ * Absolute neutrophil count \>= 1,500/mm³ * Platelet count \>= 100,000/mm³ * Serum cholesterol =\< 350 mg/dL * Triglycerides =\< 400 mg/dL * Alanine aminotransferase (ALT)/aspartate aminotransferase (AST) =\< 2.5 times upper limit of normal. * No peripheral neuropathy \> grade 2. * At least 5 years since prior chemotherapy for other types of cancer.