Cobimetinib and Atezolizumab in Treating Participants With Advanced or Refractory Rare Tumors

Inclusion Criteria: * Patients must be informed of the investigational nature of this study and must be willing to give written informed consent in accordance with institutional and federal guidelines. Patients must be able to comply with the requirements and assessments of the study protocol * Must have histologically or cytologically documented rare tumor as defined per protocol that is metastatic or locally advanced and unresectable. Patients with locally advanced cutaneous squamous cell carcinoma that are technically resectable but in whom surgery is expected to lead to substantial function impairment or disfigurement are eligible * Must be refractory or intolerant to standard lines of therapy * Must have completed prior chemotherapy, immunotherapy, or radiation therapy at least 14 days prior to start of treatment and all toxicity must be resolved to Common Terminology Criteria for Adverse Events (CTCAE) v4.0 grade 1 (with the exception of CTCAE v4.0 grade 2 neuropathy) prior to start of treatment * Presence of radiographically evaluable disease * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Tissue Parameters: a. Representative formalin-fixed paraffin-embedded (FFPE) tumor specimens in paraffin blocks (blocks are preferred) or at least 4 unstained slides, with an associated pathology report, for testing of tumor PD-L1 expression (tumor tissue from bone metastases is not evaluable for PD-L1 expression and is therefore not acceptable). b. Tumor tissue should be of good quality based on total and viable tumor content. Fine needle aspiration, brushing, cell pellet from pleural effusion, bone metastases, and lavage samples are not acceptable. For core-needle biopsy specimens, at least three cores should be submitted for evaluation. c. Patients who do not have tissue specimens meeting eligibility requirements must be willing to undergo a biopsy during the screening period * Absolute neutrophil count (ANC) \>= 1,000/mcL (obtained within 14 days prior to enrollment) * Platelets \>= 75,000/mcL (obtained within 14 days prior to enrollment) * Hemoglobin \>= 9 g/dL (obtained within 14 days prior to enrollment) * Calculated creatinine clearance \> 30 ml/min (obtained within 14 days prior to enrollment) * Aspartate aminotransferase (AST), alanine aminotransferase (ALT), and alkaline phosphatase (ALP) =\< 3 x institutional upper limit of normal (IULN) without liver mets or =\< 5 x IULN with liver metastases (obtained within 14 days prior to enrollment) * Bilirubin =\< 1.5 mg/dL (obtained within 14 days prior to enrollment) * Able to swallow pills * Negative serum pregnancy test within 7 days prior to commencement of dosing in premenopausal women. Women of non-childbearing potential may be included without serum pregnancy test if they are either surgically sterile or have been postmenopausal for \>= 1 year * Fertile men and women must use an effective method of contraception during treatment and for at least 6 months after completion of treatment as directed by their physician. Effective methods of contraception are defined as those that result in a low failure rate (i.e., less than 1% per year) when used consistently and correctly (e.g., implants, injectables, combined oral contraception or intra-uterine devices). At the discretion of the Investigator, acceptable methods of contraception may include total abstinence in cases where the lifestyle of the patient ensures compliance. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, post ovulation methods\] and withdrawal are not acceptable methods of contraception) * For individual baskets: * Appendiceal adenocarcinoma * Not considered candidate for curative surgery * Cutaneous squamous cell carcinoma * Patients with either metastatic or locally advanced cutaneous squamous cell carcinoma that are technically resectable but in whom surgery is expected to lead to substantial function impairment or disfigurement are eligible * Small bowel adenocarcinoma * Must be refractory or intolerant to at least one line of fluorouracil (5FU)-based chemotherapy for metastatic disease Exclusion Criteria: * Presence of brain metastases (unless they have been adequately treated with radiotherapy or surgery and stable for at least 30 days prior to enrollment provided patient is neurologically asymptomatic and without corticosteroid treatment for at least 7 days prior to enrollment) * Uncontrolled intercurrent illness including, but not limited to diabetes, hypertension, severe infection, severe malnutrition, unstable angina, class III-IV New York Heart Association (NYHA) congestive heart failure, ventricular arrhythmias, active ischemic heart disease, or myocardial infarction within 6 months prior to enrollment * History of or evidence of retinal pathology on ophthalmologic examination that is considered a risk factor for neurosensory retinal detachment, central serous chorioretinopathy, retinal vein occlusion (RVO), or neovascular macular degeneration * Patients will be excluded from study participation if they currently are known to have any of the following risk factors for RVO: a. Glaucoma with intraocular pressure \>= 21 mmHg b. Grade \>= 2 serum cholesterol c. Grade \>= 2 hypertriglyceridemia d. Grade \>= 2 or symptomatic hyperglycemia (fasting) e. Grade \>= 2 uncontrolled hypertension (patients with a history of hypertension controlled with anti-hypertensive medication to grade =\< 1 are eligible) * Active malignancy (other than colorectal carcinoma \[CRC\]) or a history of prior malignancy within the past 3 years. Adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, ductal carcinoma in situ, other low grade lesions such as incidental appendix carcinoid, or any other cancer from which the patient has been disease and treatment free for two years are allowed. Prostate cancer patients on active surveillance are eligible * Pregnant or nursing patients due to risk of fetal or nursing infant harm. Women/men of reproductive potential who do not agree to use an effective contraceptive method while on study and for at least 6 months after study treatment * Exclusion criteria related to study medication (any cancer immunotherapy including CD137 agonists, anti-PD-1, anti-PD-L1, or anti-CTLA4 or any MEK or ERK inhibitor) * Left ventricular ejection fraction (LVEF) \< institutional lower limit of normal or \< 50% * History or risk of autoimmune disease, including but not limited to systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjogren's syndrome, Bell's palsy, Guillain-Barre syndrome, multiple sclerosis, autoimmune thyroid disease, vasculitis, or glomerulonephritis a. Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid replacement hormone may be eligible. b. Patients with controlled type 1 diabetes mellitus on a stable insulin regimen may be eligible. c. Patients with eczema, psoriasis, lichen simplex chronicus of vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis would be excluded) are permitted provided that they meet the following conditions: * Patients with psoriasis must have a baseline ophthalmologic exam to rule out ocular manifestations * Rash must cover less than 10% of body surface area (BSA) * Disease is well controlled at baseline and only requiring low potency topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%, fluocinolone 0.01%, desonide 0.05%, alclometasone dipropionate 0.05%) * No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation \[PUVA\], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids) * History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis on screening chest computed tomography (CT) scan a. History of radiation pneumonitis in the radiation field (fibrosis) is permitted * History of human immunodeficiency virus (HIV) infection or active hepatitis B (chronic or acute) or hepatitis C infection. a. Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible. b. Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) * Active tuberculosis or severe infections within 4 weeks prior to cycle 1, day 1, including but not limited to hospitalization for complications of infection, bacteremia, or severe pneumonia * Treatment with systemic immuno-stimulatory agents (including but not limited to interferon \[IFN\] or interleukin \[IL\]-2) within 6 weeks or five half-lives of the drug (whichever is shorter) prior to cycle 1, day 1 * History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins * Prior allogeneic bone marrow transplantation or prior solid organ transplantation * For individual baskets: * Appendiceal adenocarcinoma * Must not have clinically symptomatic malignant bowel obstruction * Cutaneous squamous cell carcinoma * None * Small bowel adenocarcinoma * Must not have clinically symptomatic malignant small bowel obstruction