S1300: Pemetrexed Disodium With or Without Crizotinib in Treating Patients With Stage IV Non-Small Cell Lung Cancer That Has Progressed After Crizotinib

Inclusion Criteria: * Patients must have histologically or cytologically proven primary non-squamous non-small cell lung cancer (adenocarcinoma, large cell carcinoma, adenocarcinoma in situ, mixed histology with \< 50% squamous or unspecified); patients with tumors having squamous cell components \>= 50% are not eligible; disease must be stage IV * Patients must have documented ALK positivity at the time of initial crizotinib monotherapy using the Vysis Break-Apart FISH assay (or other Food and Drug Administration \[FDA\]-approved diagnostic test); samples are deemed to be FISH-positive if greater than or equal to 15% of scored tumor cells had split ALK 5' and 3' probe signals or had isolated 3' signal; FISH status must be documented on the Onstudy Form and a copy of the pathology report from the Vysis Break-Apart FISH assay (or other FDA-approved diagnostic test) must be submitted * Prior to registration, patients must have achieved clinical benefit with crizotinib monotherapy and subsequently have systemically progressed; clinical benefit is defined as having stable disease on crizotinib monotherapy for at least 90 days or achieving a confirmed partial or complete response; systemic progression is defined as progressive disease based on Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1, excluding progression based on brain/CNS metastases alone * Patients must have received crizotinib monotherapy at 250 mg BID on a continuous dosing schedule for at least 90 days; patients must be planning to start treatment at least three days, but no more than 30 days after discontinuing crizotinib monotherapy; patients who were not able to tolerate 250 mg BID of crizotinib are not eligible for this study * Patients must be pemetrexed-naïve; patients may have received any number of prior chemotherapy or molecularly targeted agents; if crizotinib was used in the 1st line setting then chemotherapy naive patients are also eligible; if patient received crizotinib in combination with chemotherapy, prior chemotherapy must have been discontinued at least 14 days prior to registration and all adverse events must have resolved to =\< grade 1 * Patients must have measurable disease per RECIST documented by computed tomography (CT) or magnetic resonance imaging (MRI); the CT from a combined positron emission tomography (PET)/CT may be used to document only non-measurable disease unless it is of diagnostic quality; measurable disease must be assessed within 28 days prior to registration; pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease; non-measurable disease must be assessed within 42 days prior to registration; all disease must be assessed and documented on the Baseline Tumor Assessment Form RECIST 1.1 * Patients must have a CT or MRI scan of the brain to evaluate for CNS disease within 42 days prior to registration; patient must not have brain metastases unless: (1) metastases have been treated and have remained controlled for at least 14 days following treatment or was not treated, but is asymptomatic, AND (2) patient has no residual neurological dysfunction off corticosteroids or anti-convulsants for at least 14 days * Patients may have received palliative radiotherapy to non-target lesions within 14 days prior to registration provided all radiotherapy related toxicities have resolved to =\< grade 1 prior to registration; patients must not have received any major surgery within 28 days prior to registration * Patients must not have had any prior exposure to heat shock protein (HSP)90 inhibitors (such as IPI-504 or ganetespib) or non-crizotinib ALK inhibitors (such as AP26113 or LDK378) * Patients must be offered participation in the translational medicine studies; additionally if patient has biopsy accessible disease they must be offered participation in the translational medicine studies * Absolute neutrophil count (ANC) \>= 1,500/ul * Platelet count \>= 100,000/ul * Hemoglobin \>= 9 g/dL * Serum bilirubin =\< 2 X institutional upper limit of normal (IULN) * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) or serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 2.5 x IULN * Estimated (calculated) or measured glomerular filtration rate \>= 45 mL/min (or 45 mL/min/1.73 m\^2); creatinine (mg/dl) used in calculation (Cockroft-Gault) must be obtained within 28 days prior to registration * Male patients must have free and total testosterone level obtained within 28 days prior to registration * Pre-study history and physical must be obtained with 28 days prior to registration * Patients must have Zubrod performance status 0-2 within 28 days prior to registration * Patients must be able to swallow capsules * Patients must have corrected QT (QTC) interval =\< 480 msec on electrocardiogram (EKG) at baseline; patient with congenital long QT syndrome are not eligible * No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years * Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures * REGULATORY CRITERIA: Patients or their legally authorized representative must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines * REGULATORY CRITERIA: As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system * CROSSOVER (STEP 2) REGISTRATION: Patients must have progressed systemically on Arm 2 of this study (pemetrexed monotherapy) * CROSSOVER (STEP 2) REGISTRATION: Patients must be registered to crossover (Step 2) within 30 days of discontinuing treatment on Arm 2 of this study * CROSSOVER (STEP 2) REGISTRATION: ANC \>= 1,500/ul * CROSSOVER (STEP 2) REGISTRATION: Platelet count \>= 100,000/ul * CROSSOVER (STEP 2) REGISTRATION: Serum bilirubin =\< 2 X IULN * CROSSOVER (STEP 2) REGISTRATION: SGOT (AST) or SGPT (ALT) =\< 2.5 x IULN * CROSSOVER (STEP 2) REGISTRATION: estimated (calculated) or measured glomerular filtration rate \>= 45 mL/min (or 45 mL/min/1.73 m\^2) within 28 days prior to registration; creatinine (mg/dl) used in calculation (Cockroft-Gault) must be obtained within 28 days prior to registration * CROSSOVER (STEP 2) REGISTRATION: male patients must have free and total testosterone level obtained within 28 days prior to Crossover (Step 2) Registration * CROSSOVER (STEP 2) REGISTRATION: patients must have Zubrod performance status 0-2 within 28 days prior to Crossover (Step 2) Registration