Docetaxel With or Without FGFR Inhibitor AZD4547 in Treating Patients With Recurrent Non-Small Cell Lung Cancer

Phase I: Inclusion Criteria: * All females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy * Women of childbearing potential and sexually active males must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Measurable or non-measureable disease based on Response Evaluation Criteria in Solid Tumors (RECIST) 1.1; baseline measurements and evaluations of all sites of disease must be obtained =\< 4 weeks prior to registration * Histologically or pathologically confirmed squamous NSCLC; patients whose tumors contain mixed NSCLC histologies are eligible if squamous morphology is predominant; mixed tumors with small cell anaplastic elements are not eligible * Life expectancy \>= 12 weeks * Eastern Cooperative Oncology Group (ECOG) performance status 0 - 1 * Adequate organ and marrow function * Mean resting corrected QT interval (QTc) \< 470 msec obtained from 3 consecutive electrocardiograms Exclusion Criteria: * Pregnant or breast-feeding women * Clinically important abnormalities in rhythm, conduction or morphology of resting electrocardiogram (ECG) e.g. complete left bundle branch block, third degree heart block * Factors that increase the risk of QTc prolongation or risk of arrhythmic events such as heart failure, hypokalemia, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years of age or any concomitant medication known to prolong the QT interval * Prior treatment with docetaxel (except in the adjuvant setting), or AZD4547 * Prior treatment with any other chemotherapy, immunotherapy or anticancer agents within 2 weeks prior to registration * Current evidence or previous history of retinal pigmented epithelium detachment (RPED) * Previous laser treatment or intra-ocular injection for treatment of macular degeneration * Current evidence or previous history of dry or wet age-related macular degeneration * Current evidence or previous history of retinal vein occlusion (RVO) * Current evidence or previous history of retinal degenerative diseases (e.g. hereditary) * Current evidence or previous history of any other clinically relevant chorioretinal defect * Uncontrolled brain metastases * History of allergic reactions attributed to compounds of similar chemical or biologic composition to AZD4547, docetaxel or other agents used in the study * 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 * Refractory nausea and vomiting, chronic gastrointestinal disease, inability to swallow the investigational drug, previous significant bowel resection, or any other significant gastrointestinal disorder that could, in the opinion of the Investigator, interfere with the absorption of AZD4547 * Major surgical procedure within 3 weeks prior to registration * Grade 3 or higher peripheral neuropathy, as defined by the National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE),version 4.02 * Known hepatitis B virus (HBV) or hepatitis C virus (HCV) infection * Known human immunodeficiency virus (HIV) with cluster of differentiation (CD)4 count is =\< 200 cell/mm\^3 or receiving antiretroviral therapy due to potential unfavorable interactions of the agents with the study treatment * Receiving any other investigational agents while on study * Medications that are potent inhibitors of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4), cytochrome P450, family 2, subfamily C, polypeptide 8 (CYP2C8), cytochrome P450, family 2, subfamily D, polypeptide 6 (CYP2D6), or substrates of CYP3A4 prior to the first dose of study treatment Phase II pre-registration: * Patient must have paraffin-embedded tumor specimen available for submission for determination of fibroblast growth factor receptor 1(FGFR1) amplification status Phase II Step I - Randomization: * Besides the eligibility criteria in Step I, patient must have positive tumor FGFR1 gene amplification (score FISH6) as determined by an AstraZeneca approved central laboratory. Phase II Step II: Inclusion Criteria: * Patient was randomized to docetaxel only on step 1 and progressed per RECIST v1.1 criteria; registration to step 2 must occur within 4 weeks of confirmation/determination of disease progression * Confirmed measurable disease based on RECIST 1.1; baseline measurements and evaluations of all sites of disease must be obtained =\< 4 weeks prior to registration Exclusion Criteria: * Pregnant or breast-feeding women