Paclitaxel and Carboplatin in Treating Patients With Metastatic or Recurrent Solid Tumors and HIV Infection

Inclusion Criteria: * Patients must have known HIV infection and histologically confirmed solid malignancy that is metastatic or unresectable and is therefore incurable; although the focus of this trial will be on upper aerodigestive cancers (non-small cell lung cancer, head and neck squamous cell carcinoma, and non-gastroesophageal junction esophageal cancer), patients with other incurable solid tumor with disease potentially sensitive to carboplatin and/or taxanes (including but not limited to salivary gland cancer, gastric cancer, breast cancer, ovarian cancer, or anal cancer, BUT excluding Kaposi sarcoma), will be eligible * Up to 1 prior systemic therapy regimen will be permitted for palliative treatment of metastatic or unresectable relapsed disease; however, previous chemotherapy delivered with curative-intent (i.e., chemoradiotherapy or adjuvant \[postoperative\] chemotherapy at a time disease was considered potentially curable) will be permitted; prior taxane (including paclitaxel or docetaxel) and/or platinum exposure will be permitted; however, patients must not experience disease progression within 3 months of platinum-based therapy; at least 4 weeks must have elapsed since prior chemotherapy or radiation therapy, 6 weeks if the last regimen included carmustine (BCNU) or mitomycin C; toxicities from prior anticancer therapy must have recovered to =\< Grade 1 * Serologic documentation of HIV infection at any time prior to study entry, as evidenced by positive enzyme-linked immunosorbent assay (ELISA), positive Western Blot, or any other federally approved licensed HIV test; a positive HIV viral load prior to study entry will also be permitted * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 50%) * Documented life expectancy of greater than 12 weeks * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamic-pyruvic transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal * Creatinine within normal institutional limits OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Additionally, serum magnesium and potassium must be within institutional normal limits, and a CD4 count \> 100/mcL will be required within 2 weeks of study participation * Presence of at least one measureable tumor lesion is required * Participating patients must receive medically appropriate care and treatment for HIV infection, including antiretroviral medications, when clinically indicated, and should be under the care of a physician experienced in HIV management; with the exception of treatment with zidovudine and stavudine, patients will be eligible regardless of antiretroviral regimen (no antiretroviral therapy, nonnucleoside reverse transcriptase inhibitors \[NNRTI\]-based therapy, or protease inhibitor based therapy), provided there is no intention to initiate therapy or the regimen has been stable for at least 4 weeks with no intention to change the regimen within 8 weeks following study entry; the exact regimen used for HIV therapy will be captured on Case Report Forms; as study-specific (antiretroviral therapy-based) strata fill, however, only patients fitting the remaining open strata will be accrued * Because histone deacetylase inhibitors as well as other therapeutic agents used in this trial are known to be teratogenic, 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, during the duration of study participation, and for at least 3 months following study completion; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately; women of child-bearing potential must have a negative pregnancy test within 7 days before initiation of study drug dosing; post menopausal women must be amenorrheic for at least 12 months to be considered of non-childbearing potential; (Note: A woman of childbearing potential is one who is biologically capable of becoming pregnant; this includes women who are using contraceptives or whose sexual partners are either sterile or using contraceptives) * Ability to understand and the willingness to sign a written informed consent document; as the correlative studies are critical to the clinical and scientific value of the trial, CD4 count/HIV viral load determinations will be required, and participation in the tumor-based correlative studies will be strongly recommended; additionally, investigators MUST request sample donation to the AIDS Cancer Specimen Resource (ACSR); however, the patient may refuse sample donation; patients accrued to the expansion phase of the study will be required to undergo pharmacokinetic sampling * Subjects must in the opinion of the Investigator be capable of complying with this protocol Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events (toxicities not improved to =\< Grade 1) due to agents administered more than 4 weeks earlier; additionally, patients experiencing disease progression within 3 months of platinum-based therapy will be excluded from trial participation * Due to availability of effective first- and second-line therapies (as well as disease-specific clinical trials), patients with diagnosis of active Kaposi sarcoma will be excluded from study participation; however, persons with other active malignancy with prior history of Kaposi sarcoma can be considered for participation at the discretion of the Study Chair * Patients may not be receiving any other investigational agents * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events * History of allergic reactions attributed to compounds of similar chemical or biologic composition to the study agents used in study (including hypersensitivity to paclitaxel, Cremophor, or platins) * For subjects assigned to take vorinostat, inability to take oral medications; vorinostat capsules must be administered whole; note: this criterion does NOT apply to subjects treated on the Expansion Cohort (accruals post February 1, 2013) * As ketoconazole may inhibit paclitaxel metabolism, patients receiving ketoconazole for any treatment indication are ineligible; patients receiving any other medications or substances that are inhibitors or inducers of CYP450 enzymes will be eligible; however, use all such medications or substances must be documented in the Case Report Forms * For subjects assigned to take vorinostat, prior exposure to vorinostat or other known histone deacetylase (HDAC) inhibitors for cancer therapy; patients should not have taken valproic acid, another histone deacetylase inhibitor, for at least 2 weeks prior to study enrollment; note: this criterion does NOT apply to subjects treated on the Expansion Cohort (accruals post February 1, 2013) * Since zidovudine and stavudine have potential for severe hematological toxicity potentially overlapping with toxicities of the study therapy, treatment with these agents will be disallowed * Due to potential toxicity associated with study therapy (particularly with paclitaxel), patients with peripheral neuropathy \> Grade 1 will be excluded from study participation * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, opportunistic infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements; patients with HIV infection will be eligible provided they meet the criteria specified; patients with known Hepatitis B infection should be screened for active disease prior to study participation; patients with chronic Hepatitis C infection will be eligible at the discretion of the treating investigator * Pregnant women are excluded from this study and women who become pregnant while on study must be immediately discontinued; women who are breastfeeding will not be eligible for study participation