Cediranib Maleate and Selumetinib Sulfate in Treating Patients With Solid Malignancies

Inclusion Criteria: * Histologic proof of cancer that is now considered clinically unresectable and for whom there is no standard therapy; NOTE: for the maximum tolerated dose (MTD) expansion cohort only: metastatic melanoma histology is required * Measurable and non-measurable disease are eligible * Ability to provide informed consent * Absolute neutrophil count (ANC) \>= 1500/uL (obtained =\< 21 days prior to registration) * Platelets (PLT) \>= 100,000/uL (obtained =\< 21 days prior to registration) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (obtained =\< 21 days prior to registration) * Aspartate aminotransferase (AST) =\< 2.5 x ULN or =\< 5 x ULN in presence of liver metastases (obtained =\< 21 days prior to registration) * Creatinine =\< 1.5 x ULN (obtained =\< 21 days prior to registration) * Hemoglobin (HgB) \>= 9.0 gm/dL (obtained =\< 21 days prior to registration) * Alkaline phosphatase =\< 2.5 x ULN (obtained =\< 21 days prior to registration) * Creatinine clearance \> 50 ml/min, by either Cockcroft-Gault formula or 24-hour urine collection analysis (obtained =\< 21 days prior to registration) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 * Willing to return to Mayo for follow up * Life expectancy \>= 12 weeks * Women of childbearing potential only: negative serum pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Expansion phase only: willing to provide blood samples and archived tumor tissue for correlative research purposes Exclusion Criteria: * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy * 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 * Any of the following prior therapies: * Chemotherapy =\< 28 days prior to registration * Mitomycin C/nitrosoureas =\< 42 days prior to registration * Immunotherapy =\< 28 days prior to registration * Biologic therapy =\< 28 days prior to registration * Radiation therapy =\< 28 days prior to registration * Radiation to \> 25% of bone marrow * Failure to fully recover from acute, reversible effects of prior chemotherapy regardless of interval since last treatment * Cardiac conditions as follows: * Uncontrolled hypertension (blood pressure \[BP\] \>= 150/95 despite optimal therapy) * Heart failure New York Heart Association (NYHA) class II or above or left ventricular ejection fraction \< 50% * Atrial fibrillation with heart rate \> 100 beats per minute (bpm) * Unstable ischemic heart disease (myocardial infarction \[MI\] within 6 months prior to starting treatment, or angina requiring use of nitrates more than once weekly) * Patients who require concomitant agents that prolong corrected QT interval (QTc) * Known brain or central nervous system (CNS) metastases without definitive therapy; patients who have received definitive therapy for CNS lesions may be considered if there is no evidence of progression on computed tomography (CT) or magnetic resonance imaging (MRI) imaging obtained 3 months apart * Any of the following because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration \[FDA\]-approved indication and in the context of a research investigation) * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Immunocompromised patients (other than that related to the use of corticosteroids) with the exception of patients known to be human immunodeficiency virus \[HIV\] positive and have a cluster of differentiation \[CD\]4 count \> 400 and do not require antiretroviral therapy * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy =\< 3 years prior to registration; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: if there is a history or prior malignancy, they must not be receiving other specific treatment (i.e. hormonal therapy) for their cancer * Greater than +1 proteinuria on two consecutive dipsticks taken no less than 1 week apart unless urinary protein \< 1.5g in a 24 hour (hr) period or urine protein/creatinine ratio \< 1.5 * History of exposure to AZD2171 (cediranib), AZD6244 hydrogen sulfate, or mitogen-activated protein kinase kinase (MEK), retrovirus-associated deoxyribonucleic acid (DNA) sequence (Ras) or v-RAF-1 murine leukemia viral oncogene homolog (Raf) inhibitors (sorafenib); Note: prior therapy with bevacizumab, sunitinib, pazopanib or aflibercept (vascular endothelial growth factor \[VEGF\] Trap) are allowed * Surgery within two weeks prior to registration * Significant hemorrhage (\> 30 mL bleeding/episode in previous 3 months) or hemoptysis (\> 5 mL fresh blood in previous 4 weeks) * Mean QTc interval with Bazetts correction \> 480 msec (Common Toxicity Criteria \[CTC\] grade 1) in screening electrocardiogram (ECG) or history of familial long QT syndrome * Patients who are unable to swallow tablets and capsules