Selumetinib and Cyclosporine in Treating Patients With Advanced Solid Tumors or Advanced or Metastatic Colorectal Cancer

Inclusion Criteria: * Before any study procedures are performed, subjects will have the details of the study described to them, and they will be given a written informed consent document to read; then, if subjects consent to participate in the study, they will indicate that consent by signing and dating the informed consent document in the presence of study personnel * DOSE ESCALATION PHASE: Histological or cytopathological diagnosis of an advanced cancer that is refractory to standard therapy or for which no standard therapy exists * COHORT EXPANSION PHASE: Histological or cytopathological diagnosis of advanced/metastatic unresectable colorectal cancer with known rat sarcoma viral oncogene homolog (RAS) mutational status; patients with known B-Raf proto-oncogene, serine/threonine kinase (BRAF) mutations will not be eligible for the expansion cohort; all patients must have received and progressed or be intolerant of an oxaliplatin-containing regimen and an irinotecan-containing regimen * COHORT EXPANSION PHASE: At least one tumor lesion amenable to core needle biopsy without unacceptable risk of a major procedural complication (one pretreatment and at least one on-treatment biopsy will be performed) * COHORT EXPANSION PHASE: Patient must have measurable lesions as defined by RECIST version 1.1 criteria * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Estimated life expectancy \> 3 months * Absolute neutrophil count (ANC) \>= 1,500/mcl * Platelets \>= 100,000/mcl * Hemoglobin \>= 9 g/dl * Estimated glomerular filtration rate (eGFR) greater than or equal to 60 ml/min/1.73 m\^2 * Serum total bilirubin \< 1.5 x upper limit or normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]), alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x ULN; if liver involvement, AST, ALT =\< 5.0 x ULN * Alkaline phosphatase =\< 2.5 x ULN; if liver involvement, alkaline phosphatase =\< 5.0 x ULN * Serum albumin \>= 2.5 g/dl * International normalized ratio (INR) =\< 1.5 x ULN or prothrombin time (PT) =\< 1.5 x ULN seconds above control unless patient is currently receiving warfarin therapy for the treatment or prevention of venous thrombosis * A male subject of fathering potential must use an adequate method of contraception to avoid conception throughout the study (and for up to 12 weeks after the last dose of study drug) to minimize the risk of pregnancy; if the partner is pregnant or breastfeeding, the subject must use a condom * Women of childbearing potential (WOCBP) must be using an adequate method of contraception to avoid pregnancy throughout the study and for up to 12 weeks after the last dose of study drug to minimize the risk of pregnancy; WOCBP must have a negative serum or urine pregnancy test within 72 hours before the start of the investigational product * Ability to understand and the willingness to sign a written informed consent document 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 due to agents administered more than 4 weeks earlier * Patients who are receiving any other investigational agents * Prior full field radiotherapy \< 4 weeks or limited field radiotherapy \< 2 weeks prior to first study drug administration * Patients with brain metastases may participate in this trial provided they are clinically stable; patients who are \< 1 month from definitive therapy, receiving steroid therapy or taper, or anti-convulsant medications (started for brain metastases) must not be included * History of allergic reactions attributed to compounds of similar chemical or biologic composition to AZD6244 or cyclosporine A or their excipients * Patients with a history of thrombotic or embolic events within the last six months such as a cerebrovascular accident (including transient ischemic attacks), pulmonary embolism * Cardiac conditions as follows: * Active coronary artery disease, unstable or newly diagnosed angina or myocardial infarction less than 12 months prior to first study drug administration * Class II-IV New York Heart Association (NYHA) congestive heart failure * Uncontrolled hypertension (systolic blood pressure \[BP\] \> 150 mmHg and diastolic BP \> 90 mmHg for 24 hours) despite optimal medical management; blood pressure must be below 140/90 mmHg at screening; subjects with a history of hypertension who are receiving treatment with calcium channel blockers that are cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) substrates should be changed to an alternative antihypertensive medication prior to first study drug administration * Cardiac arrhythmias requiring anti-arrhythmic therapy other than beta blockers or digoxin * Corrected QT (QTc) (Frederica) prolongation \> 480 msec * Subjects with valvular heart disease Common Terminology Criteria for Adverse Events (CTCAE) (version 4.0) grade 2 * Known left ventricular ejection fraction (LVEF) \< 50% * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris (Canadian Cardiovascular Society grade II-IV despite medical therapy), cardiac arrhythmia, active bleeding diatheses, or psychiatric illness/social situations that would limit compliance with study requirements * Known ophthalmological conditions as follows: intra-ocular pressure \> 21 mmHg, or uncontrolled glaucoma (irrespective of intra-ocular pressure); current or past history of central serous retinopathy or retinal vein occlusion * Major surgical procedure, open biopsy, or significant traumatic injury less than 3 weeks or those who receive minor surgical procedures (eg core biopsy or fine needle aspiration) within 1 week from first dose of first study drug administration * Known inability to swallow capsules * Known history of human immunodeficiency virus (HIV), hepatitis B, and/or hepatitis C (no additional laboratory tests for HIV, hepatitis \[Hep\] B, or Hep C are required for screening) * Inability to comply with study and/or follow-up procedures * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with AZD6244 * Patients with hyponatremia (sodium \< 130 mmol/L) * Baseline serum potassium \< 3.5 mmol/L (potassium supplementation may be given to restore the serum potassium above this level prior to study entry) * Baseline serum calcium \< 8.4 mg/dL (calcium supplementation may be given to restore the serum calcium above this level prior to study entry) * Prisoners or subjects who are involuntarily incarcerated * Patients who are compulsorily detained for treatment of either a psychiatric or physical (eg, infectious disease) illness