Erlotinib Hydrochloride and Cabozantinib-s-Malate Alone or in Combination as Second or Third Line Therapy in Treating Patients With Stage IV Non-small Cell Lung Cancer

Criteria: * Tumor with a sensitizing mutation in epidermal growth factor receptor (EGFR), defined as follows: * EGFR mutation testing of tumor has been performed and did not demonstrate an EGFR tyrosine kinase inhibitor sensitizing mutation; at minimum, testing for EGFR exon 19 deletion and exon 21 L858R mutations must have been included; OR * EGFR mutation testing has been attempted and is inconclusive (for example, due to lack of sufficient deoxyribonucleic acid \[DNA\] yield); OR * EGFR mutation status is unknown but tumor is positive for at least one alternative driver mutation, i.e: Kirsten rat sarcoma viral oncogene homolog (KRAS) mutation, v-raf murine sarcoma viral oncogene homolog B (BRAF) mutation, human epidermal growth factor receptor 2 (HER2) mutation, ret proto-oncogene (RET) rearrangement/fusion, or one not listed following approval by the study chair prior to registration Inclusion Criteria: * INCLUSION CRITERIA STEP 1: * Cytologically or histologically confirmed non-small cell lung carcinoma (NSCLC) * Predominant non-squamous histology (patients with NSCLC not otherwise specified \[NOS\] are eligible); mixed tumors will be categorized by the predominant cell type; if small cell elements are present the patient is ineligible * Stage IV disease (includes M1a, M1b, or recurrent disease), according to the 7th edition of the lung cancer TNM classification system * Patients must have measurable disease as defined by Response Evaluation Criteria in Solid T…