WEE1 Inhibitor MK-1775, Docetaxel, and Cisplatin Before Surgery in Treating Patients With Borderline Resectable Stage III-IVB Squamous Cell Carcinoma of the Head and Neck

Inclusion Criteria: * Provision of informed consent prior to any study specific procedures * Current diagnosis of histological or cytopathological HNSCC malignancy borderline resectable stage III up to stage IVb (T1-4, N0-2, M0) or unresectable stage IV with high nodal status defined as \>= N2b (by the American Joint Committee on Cancer \[AJCC\] 7th Edition Staging) that is amenable or appropriate for curative treatment; borderline resectability is assessed; NOTE: surgical unresectability will be defined as the combination of the treating surgeon's judgment of unresectability plus one of the following objective criteria: * Encasement of tumor or nodes to the carotid artery or 3/4 encasement of the carotid artery * Involvement of prevertebral musculature * Need for glossectomy or extensive glossal resection where functional outcome is considered unacceptable to surgeon or patient * Involvement of the cervical spine * Severe, unacceptable functional deficit that would result from any proposed definitive surgical resection * NOTE: the principal investigator (PI) of the study, Dr. Mendez, is a surgical ear, nose and throat (ENT) (head and neck) oncologist and all HNSCC cases will be discussed at the University of Washington/Seattle Cancer Care Alliance weekly tumor conference where two other ENT surgical oncologists, and co-investigators in this study, will help assess resectability; as surgical unresectability may vary from patient to patient based on individual anatomy, treating physicians may, with the approval of the surgical team, declare a tumor not meeting the above criteria to be unresectable; in this case, the reason for unresectability should be documented in the medical record; medical co-morbidity and poor performance status may not be used to declare a patient unresectable * Patients must all have available tumor tissue for biopsy and not have any bleeding diathesis and/or chronic anticoagulation that cannot be stopped for the biopsy * Eastern Cooperative Oncology Group (ECOG) 0-2 * Absolute neutrophil count (ANC) \> 1500/uL * Hemoglobin \> 9 g/dL * Platelets \> 100,000/uL * Total bilirubin within 1.5 times the institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 times ULN * Creatinine must be \< 1.5 ULN or creatinine clearance must be \> 50 mL/min (calculated by Cockcroft and Gault equation) * International normalized ratio (INR) \< 1.5 times ULN * The expanded cohort will consist of predominantly (\> 50%) p53 mutated HNSCC patients at the MTD * Willingness to use a medically acceptable method of contraception throughout the study period and for 4 weeks after the final administration of AZD1775 or longer if needed as per chemotherapies' product information (all subjects) * For female subjects with reproductive potential: a negative serum pregnancy test Exclusion Criteria: * Non-squamous cell carcinomas of the head and neck region i.e. nasopharyngeal carcinoma (World Health Organization \[WHO\] type II and III) and salivary gland carcinomas * Severe uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, unstable cardiac arrhythmia, uncontrollable hypertension or any other condition or circumstance that could interfere with adherence to the study's procedures or requirements, or otherwise compromise the study's objectives * Prior treatment with any of the chemotherapy medications (cisplatin or docetaxel) for HNSCC or with AZD1775 * Prior bone marrow transplant or history of organ transplant requiring the need for any chronic immunosuppressive medications * Prior radiation to any of the field required to treat the tumor * Any distant metastatic disease * Major psychiatric disorders which would limit compliance * Neuropathy grade 2 or higher * History of prolonged QT syndrome or electrocardiogram (ECG) at screening QT interval corrected for heart rate (QTc) of \> 470 ms with Bazett's or Fridericia's formula * Active infection requiring systemic antibiotic therapy or causing fever (temp \> 100.5 degrees Fahrenheit \[F\] or 38.1 degrees Celsius \[C\]) within 1 week prior to dosing with AZD1775 * Pregnant or breast-feeding females * Second primary malignancy within 3 years (not including in situ carcinoma of the cervix, non-melanoma skin cancer or low-grade \[Gleason score =\< 6\] localized prostate cancer) at the time of consideration for study enrollment * Known prior severe allergic/hypersensitivity to the chemotherapy or any of the components of the study treatment * Refractory nausea and vomiting, chronic gastrointestinal diseases, inability to swallow and retain the formulated oral product or previous significant bowel resection that would preclude adequate absorption of AZD1775 * Inability or unwillingness to abstain from taking any medications or herbal supplements that are moderate or strong inducers of cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) at least 1 week prior dosing with AZD1775 and while on study treatment * Pre-existing hearing impairment (patients who are willing to accept risk of further impairment will be considered after audiologic testing) * Patients taking live vaccines including yellow fever vaccinations