Patients must not have received a prior BRAF or mitogen-activated protein kinase kinase (MEK) inhibitor
Any number and type of prior anticancer therapies are allowed except BRAF or mitogen-activated protein kinase kinase (MEK) inhibitors
Prior treatment with a mitogen activated-protein kinase (MAPK) inhibitor
History of prior significant toxicity (Grade  or higher that required permanent treatment discontinuation) from a BRAF, MEK (MAPK [Mitogen-activated protein]/ERK kinase) or ERK inhibitor.
No prior therapy with inhibitors affecting the mitogen-activated protein kinase (MAPK) pathways at any level (BRAF, mitogen-activated protein [MAP]/extracellular signal-related kinase [ERK] kinase [MEK], neuroblastoma RAS viral [v-ras] oncogene homolog [NRAS], ERK inhibitors) for unresectable stage IIIC or stage IV (metastatic) melanoma; no limit to other therapies (immunotherapy or chemotherapy); prior systemic treatment in the adjuvant setting is allowed; (note: ipilimumab treatment must end at least  weeks prior to study day )
Prior therapy with compounds targeting PD-, PD-L, BRAF, mitogen-activated protein kinase (MEK) or other molecules in the mitogen-activated protein kinase (MAPK) pathway
Previous treatment with cobimetinib or any other mitogen-activated protein/extracellular signal-regulated kinase (MEK) inhibitor
Prior treatment with clusters of differentiation (CD)  (CD) agonists or immune checkpoint blockage therapies, anti-programmed death protein-, anti-program death-ligand , mitogen-activated protein kinase (MEK) inhibitor
Part B: Have advanced/metastatic cancer carrying activating mitogen-activated protein kinase (MAPK) pathway alteration
Prior therapy with a BRAF and/or mitogen-activated protein kinase kinase (MEK) and/or extracellular signal-regulated kinase (ERK) inhibitors
Prior exposure to experimental treatment targeting rapidly accelerated fibrosarcoma (Raf), MAP kinase/ERK kinase (MEK), or the mitogen-activated protein kinase (MAPK) pathway
Previous mitogen-activated protein kinase kinase (MEK) inhibitor use such as PD-; CI; AS; GDC ; ARRY; GSK
Prior treatment with a MAP-kinase pathway inhibitor (RAS, RAF, ERK, MEK)
Prior therapy with selumetinib or another specific mitogen-activated protein kinase kinase (MEK) inhibitor is not permitted
Prior treatment with a mitogen-activated protein kinase kinase (MEK) inhibitor
Use of other investigational drugs within  days (or five half-lives, whichever is shorter; with a minimum of  days from the last dose) preceding the first dose of trametinib and during the study; patients previously treated with v-raf murine sarcoma (RAF) and/or mitogen-activated protein kinase (MEK) inhibitors are excluded from the study; multikinase antiangiogenic tyrosine kinase inhibitors such as regorafenib, sorafenib, sunitinib, etc. whose primary mechanism of action is not RAF inhibition, are allowed; if there are any questions, please contact study's principal investigator
Prior BRAF or mitogen-activated protein kinase kinase (MEK) directed therapy; patients who have received prior interferon are eligible
Has had prior systemic cancer therapy within the past four weeks or v-raf murine sarcoma viral oncogene homolog B (B-RAF) or mitogen-activated protein kinase (MEK) inhibitors within  days at the time of the start of the lymphodepletion regimen (Turnstile II)
Prior exposure to mitogen-activated protein kinase kinase (MEK), RAS, or RAF inhibitors (note: previous exposure to sorafenib is allowed) OR history of hypersensitivity to selumetinib, thyrotropin alpha (Thyrogen), or any excipient agents
Has had prior B-RAF or mitogen-activated protein kinase kinase (MEK) targeted therapy within  days prior to the start of lymphodepletion regimen (Cohort A and Cohort B)
Part : Has received prior therapy with cancer vaccines, or compounds targeting programmed cell death ligand  (PD-L), PD-L, cytotoxic T-lymphocyte-associated protein  (CTLA-), or Mitogen-activated protein kinase (MAPK)/Extracellular signal-regulated Kinase (MEK).
Must have progressed on at least one prior line of platinum-based therapy for stage IV NSCLC (excluding docetaxel or mitogen activated protein kinase kinase [MEK] inhibitors)
Melanoma patients must be intolerant of, or have disease that has proven refractory to approved therapies such as B-Raf proto-oncogene, serine/threonine kinase (BRAF) or mitogen-activated protein kinase kinase  (MEK) inhibitors for BRAF-positive metastatic melanoma and/or checkpoint blockade with either anti-programmed cell death  (PD) or anti-cytotoxic T-lymphocyte-associated protein  (CTLA-) for metastatic melanoma
Prior exposure to Bcl- inhibitors, murine double minute  (MDM) antagonists or prior exposure to experimental treatment targeting Raf, mitogen-activated protein kinase (MEK), or the mitogen-activated protein kinase (MAPK) RAS/RAF/MEK/ERK MAPK pathway
TREATMENT: Patients who have received prior mitogen-activated protein kinase kinase (MEK) inhibitors would not be eligible to receive trametinib DMSO on study; if these patients have mutations of interest in pathways other than the rat sarcoma (RAS) pathway, they will be eligible to receive agents based on that mutation
Use of other investigational drugs within  days (or five half-lives, whichever is shorter; with a minimum of  days from the last dose) preceding the first dose of study treatment and during the study; patients that have used other BRAF or mitogen-activated protein kinase kinase (MEK) inhibitor are excluded
Prior chemotherapy, including targeted therapy such as v-raf murine sarcoma viral oncogene homolog B (BRAF) or mitogen-activated protein kinase kinase (MEK) inhibition
Prior use of raf-kinase inhibitors (sorafenib), vascular endothelial growth factor (VEGF) inhibitors, mitogen-activated protein kinase (MAPK)/extracellular-signal-related kinase (ERK) kinase (MEK) inhibitors, or farnesyl transferase inhibitors
PRIOR/CONCURRENT THERAPY CRITERIA: Patients must not have received prior docetaxel; patients must not have received therapy with a drug known to be either a mitogen-activated protein kinase (MEK) inhibitor or a phosphatidylinositol  kinase (PIK)/v-akt murine thymoma viral oncogene homolog  (AKT)/mammalian target of rapamycin (mTOR) pathway inhibitor
Receipt of prior mitogen-activated protein kinase inhibitor pathway agents including mitogen-activated protein kinase inhibitor and BRAF kinase inhibitor
Prior treatment with a mitogen-activated protein kinase kinase (MEK) inhibitor of any kind
History of prior significant toxicity from another mitogen-activated protein kinase (MEK) pathway inhibitor requiring discontinuation of treatment
Prior treatment with a Mitogen-Activated protein Kinase (MEK) inhibitor or any docetaxel-containing regimen (prior treatment with paclitaxel is acceptable).
Any number and type of prior anticancer therapies except BRAF or mitogen-activated protein kinase (MEK) inhibitors
Prior treatment with a MEK (Mitogen-Activated Protein Kinase) inhibitor or any docetaxel-containing regimen (prior treatment with paclitaxel is acceptable)
Prior therapy with ipilimumab, other BRAF inhibitors, or mitogen-activated protein kinase (MEK) inhibitors
Prior therapy with a mitogen-activated protein kinase kinase (MEK)-inhibitor
SELUMETINIB ARM: Any prior exposure to mitogen-activated protein kinase kinase (MEK), rat sarcoma (Ras), or v-raf- murine leukemia viral oncogene homolog  (Raf) inhibitors
Patients with known concurrent activating retrovirus-associated DNA sequence (RAS)/v-RAF- murine leukemia viral oncogene homolog (RAF) mutation or loss of function mutation or deletion in neurofibromin (NF) of NF resulting in mitogen-activated protein (MAP) kinase pathway activation; patients are not required to be evaluated for these alterations if not already performed
Prior treatment with any BRAF inhibitor or any mitogen-activated protein/extracellular signal-regulated kinase inhibitor.
Prior therapy (except for adjuvant immunotherapy) with a BRAF and/or mitogen-activated protein kinase kinase (MEK) and/or extracellular signal-regulated kinase (ERK) inhibitors
The subject has been previously treated with a Mitogen-activated protein/extracellular signal-regulated kinase (MEK) inhibitor.
Part  ONLY: Previous treatment with dabrafenib, another RAF inhibitor, or a mitogen-activated protein kinase (MEK) inhibitor (exception: prior treatment with sorafenib is permitted).
Prior treatment with selumetinib or another specific mitogen-activated protein kinase (MEK)/ inhibitor (unless the subject meets criteria for re-treatment)
No prior therapy with a BRAF inhibitor or mitogen-activated protein kinase kinase (MEK) inhibitor or leflunomide
At least  days from prior treatment (including adjuvant interferon) except in the case of a v-raf murine sarcoma viral oncogene homolog B (BRAF) inhibitor (such as vemurafenib); concurrent treatment with a BRAF inhibitor +/- mitogen activate protein kinase kinase (MEK) inhibitor is permitted; concurrent treatment with an anti-programmed death- (PD) agent is permitted
Prior treatment with a BRAF or mitogen-activated protein kinase kinase (MEK) inhibitor for metastatic melanoma (treatment in the adjuvant setting is allowed)
Additional Inclusion Requirements for TAK- + nivolumab a) BRAF V mutation-positive or NRAS mutation-positive disease previously untreated with RAF, MEK, or other inhibitors of the mitogen-activated protein kinase (MAPK) pathway. Participants who have progressed on these agents can still be enrolled in TAK- (plozalizumab) + nivolumab or vedolizumab + nivolumab + ipilimumab.