Atezolizumab With Chemotherapy in Treating Patients With Anaplastic or Poorly Differentiated Thyroid Cancer

Inclusion Criteria: * Histologically confirmed anaplastic thyroid or poorly differentiated thyroid carcinomas. * Patients deemed to have unresectable locoregional disease or metastatic disease. Patients who are unwilling to undergo surgery or external beam radiation are also eligible. * Patients with poorly differentiated thyroid cancer must have at least one target lesion by RECIST version 1.1. This is not a requirement for ATC patients. * Total bilirubin =\< 1.5 x upper limit of normal (ULN). Total bilirubin: 3 x ULN for patients with Gilbert's syndrome. * Aspartate aminotransferase (AST) serum glutamic-oxaloacetic transaminase (SGOT) / alanine aminotransferase (ALT) serum glutamate pyruvate transaminase (SGPT) =\< 2.5 x ULN, (5 x ULN for patients with concurrent liver metastases). * Serum creatinine =\< within 1.5 x ULN. * Absolute neutrophil count (ANC) \>= 1.0 x 10\^9 /L. * Platelets (PLT) \>= 100 x 10\^9 /L. * For patients receiving therapeutic anticoagulation: stable anticoagulant regimen and stable international normalized ratio (INR) during the 28 days immediately preceding initiation of study treatment. * Subjects must be willing to undergo tumor biopsy prior to and after treatment with atezolizumab, unless in the opinion of the treating physician, a biopsy is not feasible or safe. * Eastern Cooperative Oncology Group (ECOG) performance score (PS) =\< 2. * Age and reproductive status: * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test within 14 days prior to the start of study drug and must use effective contraceptives throughout the duration of the study. Males who are sexually active with WOCBP must agree to use effective contraception throughout the duration of the study. Azoospermic males and WOCBP who are continuously not heterosexually active are exempt from contraceptive requirements. * A Women of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) and is not postmenopausal. Menopause is defined as 12 months of amenorrhea in a woman over age 45 years in the absence of other biological or physiological causes. * Ability to provide informed consent. * ADDITIONAL INCLUSION CRITERIA FOR BRAF MUTATION (COHORT 1): Patients with a BRAFV600E mutation being considered for the triplet combination (vemurafenib + cobimetinib + atezolizumab) must meet the following end organ function criteria: * COHORT 1: ANC \>= 1.5 × 10\^9 /L without granulocyte colony-stimulating factor support. * COHORT 1: White blood cell (WBC) count \>= 2.5 x 10\^9 /L. * COHORT 1: Lymphocyte count \>= 0.5 x 10\^9/L. * COHORT 1: Platelet count \>= 100 × 10\^9 /L without transfusion. * COHORT 1: Hemoglobin \>= 9.0 g/L without transfusion. * COHORT 1: Serum albumin \> =2.5 g/L * COHORT 1: Total bilirubin =\< 1.5 x ULN * COHORT 1: AST and ALT =\< 2.0 x ULN * COHORT 1: Alkaline phosphatase (ALP) =\< 2.5 x ULN or, for patients with documented liver or bone metastases, ALP =\< 5 x ULN * COHORT 1: Serum creatinine =\< 1.5 x ULN or creatinine clearance (CrCl) \>= 40 mL/min on the basis of measured CrCl from a 24-hour urine collection or Cockcroft-Gault glomerular filtration rate estimation. * COHORT 1: Patients with BRAF mutation may be screened for eligibility in cohorts 2, 3, or 4 (in this order of preference) if they do not meet the entry criteria for cohort 1. Exclusion Criteria: * Subjects with an active, known or suspected autoimmune disease. Subjects with type I diabetes mellitus on stable insulin regimen, hypothyroidism only requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll. * For patients not receiving therapeutic anticoagulation: INR or partial thromboplastin time (aPTT) \> 1.5 x ULN within 28 days prior to initiation of study treatment. * Prior treatment with anti-PD-1, or anti-PD-L1 therapeutic antibody or pathway targeting agents. Patients who have received prior treatment with anti-CTLA-4 may be enrolled, provided the following requirements are met: Minimum of 12 weeks from the first dose of anti-CTLA-4 and \> 6 weeks from the last dose, and no history of severe immune-related adverse effects from anti-CTLA 4 (National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] grade 3 and 4). * Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease. * History of human immunodeficiency virus (HIV) infection or active hepatitis B (chronic or acute) or hepatitis C infection 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. However, patients with past or resolved hepatitis B virus (HBV) should be monitored for reactivation by a specialist. Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA). * Pregnant or lactating women. All pre-menopausal women being screened must have a negative serum pregnancy test within 14 days prior to commencement of dosing. Women of non-childbearing potential may be included 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. * Untreated brain metastases. * Chemotherapy within 21 days of enrollment with the exception of paclitaxel or nab-paclitaxel (Abraxane). Patients who have received one course of these agents prior to study entry are eligible. (One course of weekly paclitaxel or nab-paclitaxel is 3 doses. One course of every 3 week dosing of paclitaxel or nab-paclitaxel is 1 dose). Patients who have received prior radiosensitizing chemotherapy are eligible. * The use of corticosteroids is not allowed for 10 days prior to initiation of atezolizumab except patients who are taking steroids for physiological replacement. Inhaled or topical steroids, and adrenal replacement steroid doses are permitted in the absence of active autoimmune disease. This does not apply to patients receiving steroids as pre-medications for paclitaxel administration. * Grade \>= 2 uncontrolled hypertension (patients with a history of hypertension controlled with anti-hypertensive medication to Grade =\< 1 are eligible). * ADDITIONAL EXCLUSION CRITERIA FOR non-BRAF/non-RAS MUTATION (COHORT 3): * Patients with clinically significant hemoptysis or tumor bleeding within two weeks prior to first dose of targeted therapy. * Patients with suspected tracheal or esophageal invasion are excluded from cohort 3 due to the high risk of trachealesophageal fistula. Patients excluded from cohort 3 may be enrolled on taxane + atezolizumab cohort (cohort 4). * ADDITIONAL EXCLUSION CRITERIA FOR COHORTS 1 and 2: Ocular Exclusion Criteria for vemurafenib and cobimetinib containing cohorts-cohorts 1 and 2. (However, these patients may be assigned other cohorts if they do not meet the ocular exclusion criteria): 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 participation in cohorts 1 and 2 if they currently are known to have any of the following risk factors for RVO unless a retinal specialist has determined that the risk of retinal detachment is low: * History of serous retinopathy. * History of retinal vein occlusion. * History of ongoing serous retinopathy or RVO at baseline. * ADDITIONAL EXCLUSION CRITERIA FOR PATIENTS IN COHORT 1 (vemurafenib+cobimetinib+atezolizumab): History of clinically significant cardiac dysfunction, including the following: * Mean (average of triplicate measurements) QTc interval corrected using Fridericia's method \>= 480 ms at screening, or uncorrectable abnormalities in serum electrolytes (sodium, potassium, calcium, magnesium, and phosphorus). * ADDITIONAL EXCLUSION CRITERIA FOR PATIENTS IN COHORTS 1 and 2 (cobimetinib-containing cohorts): * Unstable angina, or new-onset angina within 3 months prior to initiation of study treatment. * Symptomatic congestive heart failure, defined as New York Heart Association Class II or higher. * Myocardial infarction within 3 months prior to initiation of study treatment. * Left ventricular ejection fraction below the institutional lower limit of normal or below 50%, whichever is lower.