Tivantinib and Topotecan Hydrochloride in Treating Patients With Advanced or Metastatic Solid Tumors

Inclusion Criteria: * Patients must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective; for the expansion group, patients must have histologically or cytologically confirmed small cell lung cancer previously treated with one or more chemotherapy or chemoradiotherapy regimens, at least one of which must have been platinum-based * As of amendment dated 01/24/2014 the study will not be pursuing an expansion cohort in small cell lung cancer * Karnofsky \>= 60% * Life expectancy of greater than 12 weeks * Hemoglobin \>= 9.0 g/dL * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 X institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal (=\< 5 X institutional upper limit of normal if the rise can be attributed to liver metastases) * Serum creatinine =\< 1.5 X institutional upper limit of normal OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Women of child-bearing potential and men 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 she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of ARQ 197 administration * 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 * Patients with untreated brain metastases should be excluded from this clinical trial; however patients with stable brain disease (off corticosteroids) at least 2 weeks after completion of appropriate therapy for their brain metastases are eligible; patients who require enzyme-inducing anti-convulsants (EIAC) should be switched to non-EIAC and be on a stable dose of the new agent for at least 2 weeks prior to treatment on this protocol * History of allergic reactions attributed to compounds of similar chemical or biologic composition to ARQ 197 or topotecan * The metabolism and consequently overall pharmacokinetics of ARQ 197 could be altered by inhibitors and/or inducers or other substrates of cytochrome P450 2C19 (CYP2C19) and cytochrome P450 3A4 (CYP3A4); while inhibitors/inducers of these cytochrome P450 isoenzymes are not specifically excluded, investigators should be aware that ARQ 197 exposure may be altered by the concomitant administration of these drugs; caution should be applied when CYP2C19 inhibitors such as omeprazole, fluvoxamine, fluconazole, ticlopidine, rabeprazole, fluoxetine, and moclobemide, or strong CYP3A4 inhibitors such as atazanavir, clarithromycin, indinavir, itraconazole, ketoconazole, nefazodone, nelfinavir, ritonavir, saquinavir, ciprofloxacin, telithromycin, troleandomycin (TAO), or voriconazole, are used as concomitant therapy; because the lists of these agents are changing, it is important to consult an updated list; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * History of congestive heart failure defined as class II to IV per New York Heart Association (NYHA) classification; active coronary artery disease (CAD); clinically significant bradycardia or other uncontrolled, cardiac arrhythmia defined as \>= grade 3 according to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE), version 4.0, or uncontrolled hypertension; myocardial infarction occurring within 6 months prior to study entry (myocardial infarction occurring \> 6 months prior to study entry is permitted) * 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 * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with ARQ 197 * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible * Patients previously treated with topotecan are ineligible * Patients previously treated with ARQ 197 are ineligible * Patients unable to swallow ARQ 197 pills are ineligible