Anetumab Ravtansine and Atezolizumab in Treating Participants With Advanced Non-small Cell Lung Cancer

Inclusion Criteria: * PRE REGISTRATION ? INCLUSION CRITERIA: Ability to understand and the willingness to sign a written informed consent document * PRE REGISTRATION ? INCLUSION CRITERIA: Patient has disease amenable to biopsy if the archival tissue sample is unavailable; note: Archive sample must not be older than 12 months * REGISTRATION ? INCLUSION CRITERIA * Phase I only: Diagnosis of advanced/metastatic NSCLC for which no standard treatment option; Phase II only: Advanced NSCLC patients who have received at least 1 platinum-based systemic chemotherapy regimen * Patients with tumors having actionable genomic alterations should have received prior therapy with Food and Drug Administration (FDA) approved agents targeting these aberrations (ie EGFR, ALK, ROS1, BRAF V600E) * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Phase II only: Must have at least one measurable lesion as defined by Response Evaluation Criteria in Solid Tumors (RECIST) criteria * Ability to understand and the willingness to sign a written informed consent document * Histological or cytologically confirmed NSCLC that shows moderate or stronger mesothelin expression in 30% of tumor cells by a companion assay; MSLN expression score using Ventana immunohistochemistry (IHC) SP74 assay; Phase I only: In addition 5- 30% tumor cells and 1, 2, or 3+ MSLN score; Phase II only: 30% tumor cells and either 2+/3+ * Life expectancy of \>= 12 weeks * Absolute neutrophil count \>= 1.5 ? 10\^9/L …