Binimetinib With Docetaxel in Treating Patients With Previously Treated, Stage IV Non-small Cell Lung Cancer

Inclusion Criteria: * Signed written informed consent * A histologically or cytologically confirmed diagnosis of stage IV NSCLC * Must have progressed on at least one prior line of platinum-based therapy for stage IV NSCLC (excluding docetaxel or mitogen activated protein kinase kinase \[MEK\] inhibitors) * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Measurable disease defined by RECIST criteria * Ability to provide adequate tissue from archival tumor specimen; confirmation of adequate tissue is required prior to enrollment * In expansion cohort only: patient's kirsten rat sarcoma viral oncogene homolog (KRAS) mutational status must be known prior to enrollment * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Hemoglobin (Hgb) \>= 9 g/dL without transfusions * Platelets (PLT) \>= 100 x 10\^9/L without transfusions * Aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) =\< 1.5 x upper limit of normal (ULN) * Total bilirubin =\< 1.5 x ULN * Creatinine =\< 1.5 mg/dL * Left ventricular ejection fraction (LVEF) \>= 45% as determined by a multigated acquisition (MUGA) scan or echocardiogram, and QTc interval =\< 480ms * Able to take and retain oral medications * Patient is deemed by the investigator to have the initiative and means to be compliant with the protocol (treatment and follow-up) * Negative serum beta-human chorionic gonadotropin (HCG) test (female patient of childbearing potential only) performed locally within 72 hours (hrs) prior to first dose Exclusion Criteria: * History or current evidence of retinal vein occlusion (RVO) or current risk factors to RVO (e.g. uncontrolled glaucoma or ocular hypertension, history of hyperviscosity or hypercoagulability syndromes) * History of Gilbert syndrome * Previous or concurrent malignancy within the last 3 years with the exception of adequately treated skin basal or squamous cell with adequate wound healing * Prior therapy with a MEK-inhibitor or docetaxel for metastatic non-small cell lung cancer (docetaxel in the adjuvant setting will be allowed) * Epidermal growth factor receptor (EGFR) mutant and/or anaplastic lymphoma receptor tyrosine kinase (ALK) gene rearranged tumor * Impaired cardiovascular function or clinically significant cardiovascular diseases, including any of the following within 6 months of enrollment: myocardial infarction, unstable angina, coronary artery bypass graft (CABG), coronary angioplasty, or stenting, symptomatic chronic heart failure, clinically significant cardiac arrhythmias and/or conduction (except atrial fibrillation and paroxysmal supraventricular tachycardia) * Uncontrolled arterial hypertension despite appropriate medical therapy * Known positive serology for human immunodeficiency virus (HIV), active hepatitis B, and/or active hepatitis C infection * Neuromuscular disorders that are associated with elevated creatine kinase (CK) (e.g., inflammatory myopathies, muscular dystrophy) * Subjects planning on embarking on a new strenuous exercise regimen after first dose of study treatment that can result in significant increases in plasma CK levels * Impairment of gastrointestinal function or gastrointestinal disease (e.g., ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption) * Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to concerns regarding safety or compliance with clinical study procedures * Patients who have undergone major surgery =\< 3 weeks prior to starting study drug or who have not recovered from side effects of such procedure * Pregnant or lactating women * Women of child-bearing potential defined as all women physiologically capable of becoming pregnant who are unwilling to use highly effective methods of contraception throughout the study and 15 days after study drug discontinuation * Sexually active males unless they are willing to use a condom during intercourse while taking the drug and for 15 days after stopping treatment and are willing to forego fathering a child in this period; a condom is required to be used also by vasectomized men * Medical, psychiatric, cognitive or other conditions that may compromise the patient's ability to understand the patient information, give informed consent, comply with the study protocol or complete the study * Grade II or greater neuropathy at baseline * Symptomatic brain metastases (if a patient has brain metastases and is on steroids, the steroid dose must have been stable for at least 14 days) or brain metastases that have not been previously treated * Treatment with prior chemotherapy, monoclonal antibodies, other protein or peptide therapeutics or anticancer immunotherapy within 21 days of the first dose of study drug * Treatment with prior radiotherapy within 28 days of initiating study drug; however, if the radiation portal covered =\< 10% of the bone marrow reserve, the patient may be enrolled without respect to the end date of radiotherapy