* Have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 at the time of study treatment initiation * Have pathologically confirmed diagnosis of NSCLC (Phase I, Phase II Studies A, C and D, E and Expansion Cohort) or squamous cell head and neck cancer (Phase II Study B) * Must be eligible for treatment with nivolumab as standard of care (for nivolumab treatment groups only) * Phase II Study A and Expansion Cohort AE: Patients with advanced (metastatic) NSCLC, whose disease progressed during or after platinum based therapy * Phase II Study B: Patients with advanced recurrent head and neck squamous cell carcinoma * Phase II Study C: Patients with unresectable NSCLC with PD-L1 expression \>= 50% for first line therapy in advanced stage. In the rare event that there is a discrepancy in the results of PD-L1 testing (i.e. 2 or more specimens were tested, etc.), eligibility status will be per the discretion of the principal investigator (PI) after review of other available biomarker testing * Phase II Study D: Patients with advanced squamous NSCLC with PD-L1 expression \<50% with PR/CR or stable disease by RECIST after at least 4 cycles of induction chemoimmunotherapy with platinum-based combination with pembrolizumab, prior to initiation of maintenance pembrolizumab * Phase II Study E: Patients with advanced NON-Squamous NSCLC (without EGFR/ALK/ROS-1/KRAS mutations) with PD-L1 expression \<50% with PR/CR or stable disease by RECIST after at least 4 cycles of induction chemoimmunotherapy with platinum-based combination with pembrolizumab, prior to initiation of maintenance pembrolizumab * NSCLC patients in study A and expansion cohort AE with EGFR or ALK genomic tumor aberrations (determined through either tissue- or liquid biopsy-based platform) should have disease progression on Food and Drug administration (FDA)-approved therapy for these aberrations prior to receiving nivolumabanti-PD1 therapy; patients with smoking history being considered for Study C may enroll and be treated pending results of molecular testing * Have at least 6 month life expectancy * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Platelets \>= 100 x 10\^9/L * Hemoglobin \>= 9 g/dL * Serum /plasma creatinine =\< 1.5 x institution upper limit of normal (ULN) or estimated glomerular filtration rate (GFR) (measured or calculated with Cockcroft and Gault formula) \> 45 ml/min * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN (ALT and AST =\< 5 x ULN is acceptable if liver metastases are present) * Total serum bilirubin =\< 1.5 x ULN; for patients with well documented Gilbert's syndrome, total bilirubin =\< 3 x ULN with direct bilirubin within normal range * Troponin-I =\< ULN and B-type natriuretic peptide (BNP) \< 200 pg/ml * Left ventricular ejection fraction (LVEF) \>= lower limit of normal (LLN) (institutional limit) * Patients enrolled onto Phase I dose escalation or Expansion Cohort (AE) must have presence of evaluable disease; patients enrolled onto Phase II studies A, B or C must have measurable disease as defined in RECIST 1.1. Patients enrolled onto Phase II D or E must have measurable disease as defined in RECIST 1.1 prior to induction chemmoimmunotherapy. * Participants of child-bearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry; 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 * Participant or legal representative must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure * Phase II studies: Participant agrees to provide tumor biopsy tissue before treatment, blood samples at the start of treatment and at multiple times during the study and, a tumor biopsy at the end of the trial or after disease progression; archival formalin-fixed paraffin-embedded (FFPE) tissue is permitted for Expansion Cohort AE and Cohort D or E (i.e.; fresh biopsy is NOT mandatory). Archival FFPE tissue is also permitted for Study C patients provided that tissue is adequate and no systemic anti-cancer therapy had been administered between the time specimen was obtained and start of protocol therapy