Selinexor and Docetaxel in Treating Patients With Recurrent or Metastatic Squamous Cell Lung Cancer

Inclusion Criteria: * Written informed consent in accordance with federal, local, and institutional guidelines * Patients with recurrent or metastatic squamous cell carcinoma of the lung - diagnosis must be histologically confirmed * Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST 1.1) at the time of study entry * Objective evidence of disease progression on study entry * Prior systemic anticancer therapy: Patients will have received at least 1 platinum-based chemotherapy regimen, but no more than 2 cytotoxic chemotherapy regimens in the setting of recurrent or metastatic disease; the regimen(s) may have included biological, molecularly targeted or immune therapies; adjuvant chemotherapy is considered 1 cytotoxic chemotherapy regimen if the last administration occurred \< 1 year prior to entry * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Absolute neutrophil count (ANC) \> 1500/mm\^3 * Platelets count \> 100,000 mm\^3 and less than 1,000,000 mm\^3 * Total bilirubin \< 2 times the upper limit of normal (ULN) (except patients with Gilbert's syndrome who must have a total bilirubin of \< 3 times ULN) * Alanine aminotransferase (ALT) \< 2.5 times ULN; in the case of known (radiological and/or biopsy documented) liver metastasis, ALT \< 5.0 times ULN is acceptable; patients with \> 3 liver metastases at enrollment will be excluded * Estimated creatinine clearance of \>= 30 mL/min, calculated using the formula of Cockcroft and Gault * Amylase =\< 1.5 x ULN * Lipase =\< 1.5 x ULN * Alkaline phosphatase limit =\< 2.5 x ULN * Female patients of child-bearing potential must agree to use dual methods of contraception and have a negative serum pregnancy test at screening; male patients must use an effective barrier method of contraception if sexually active with a female of child-bearing potential throughout the study and for three months following the last dose of selinexor * Resolution to grade =\< 1 by National Cancer Institute Common Terminology Criteria for Adverse Events version 4.03 (CTCAE v4.03) of all clinically significant toxic effects of prior anti-cancer therapy (with the exception neuropathy, which may be =\< grade 2 within 14 days prior to cycle 1 day 1) Exclusion Criteria: * Patients who are pregnant or lactating * Radiation, chemotherapy, or immunotherapy or any other anticancer therapy =\< 2 weeks prior to cycle 1 day 1 * Prior treatment with selective inhibitor of nuclear export (SINE) inhibitor * Major surgery within four weeks before cycle 1, day 1 * Unstable cardiovascular function: * Electrocardiography (ECG) abnormalities requiring treatment, or * Congestive heart failure (CHF) of New York Heart Association (NYHA) class \>= 3 * Myocardial infarction (MI) within 3 months * Uncontrolled infection requiring parenteral antibiotics, antivirals, or antifungals within one week prior to first dose; patients with controlled infection or on prophylactic antibiotics are permitted in the study * Known to be human immunodeficiency virus (HIV) seropositive * Known active hepatitis A, B, or C infection; or known to be positive for hepatitis C virus (HCV) RNA or hepatitis B surface antigen (HBsAg) (hepatitis B virus \[HBV\] surface antigen) * Any underlying condition that would significantly interfere with the absorption of an oral medication * Patients with markedly decreased visual acuity * Serious psychiatric or medical conditions that could interfere with treatment * Participation in an investigational anti-cancer study =\< 3 weeks prior to cycle day 1 * Concurrent therapy with approved or investigational anticancer therapeutic other than steroids * Patients with coagulation problems and active bleeding in the last month (peptic ulcer, epistaxis, spontaneous bleeding) * Uncontrolled brain metastases; patients with brain metastases are permitted if they have received appropriate therapy and demonstrated control of the brain metastases following therapy; patients with known brain metastases will require magnetic resonance imaging (MRI) brain to demonstrate disease control prior to enrollment (lack of symptom progression for two weeks off therapeutic doses of steroids, excluding chronic steroids used for control of chronic obstructive pulmonary disease \[COPD\]) * Renal failure requiring hemodialysis or peritoneal dialysis * Patients with significantly diseased or obstructed gastrointestinal tract, malabsorption, uncontrolled vomiting or diarrhea or inability to swallow oral medications * Patients who are severely underweight in the opinion of the investigator * Prior cancer diagnosis is allowed if patient is disease-free for \>= 3 years, or disease free for \< 3 years for treated basal cell/squamous cell skin cancer or in situ cervical cancer