Transoral Surgery Followed By Low-Dose or Standard-Dose Radiation Therapy With or Without Chemotherapy in Treating Patients With HPV Positive Stage III-IVA Oropharyngeal Cancer

Inclusion Criteria:

Registration to Surgery (Arm S)

* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* Patients must have newly diagnosed, histologically or cytologically confirmed squamous cell carcinoma or undifferentiated carcinoma of the oropharynx; patients must have been determined to have resectable oropharyngeal disease; patients with primary tumor or nodal metastasis fixed to the carotid artery, skull base or cervical spine are not eligible
* Patients must have American Joint Committee on Cancer (AJCC) TNM tumor stage III, IV a, or IV b (with no evidence of distant metastases) as determined by imaging studies (performed \< 30 days prior to pre-registration) and complete head and neck exam; the following imaging is required: computed tomography (CT) scan with IV contrast or magnetic resonance imaging (MRI)
* Patients must have biopsy-proven p16+ oropharynx cancer; the histologic evidence of invasive squamous cell carcinoma may have been obtained from the primary tumor or metastatic lymph node. It is required that patients have a positive p16 IHC (as surrogate for HPV) status from either the primary tumor or metastatic lymph node.
* Carcinoma of the oropharynx associated with HPV as determined by p16 protein expression using immunohistochemistry (IHC) performed by a Clinical Laboratory Improvement Amendments (CLIA) approved laboratory; using p16 antibody obtained from Roche mtm laboratories AG (CINtec, clone E6H4) is recommended
* Patients with a history of a curatively treated malignancy must be disease-free for at least two years except for carcinoma in situ of cervix and/or non-melanomatous skin cancer
* Patients with the following within the last 6 months prior to pre-registration must be evaluated by a cardiologist and/or neurologist prior to entry into the study

  * Congestive heart failure \> NYHA Class II
  * Cerebrovascular accident (CVA)/transient ischaemic attack (TIA)
  * Unstable angina
  * Myocardial infarction
* Absolute neutrophil count \>= 1,500/mm\^3
* Platelets \>= 100,000/mm\^3
* Total bilirubin =\< the upper limit of normal (ULN)
* Calculated creatinine clearance must be \> 60 ml/min using the Cockcroft-Gault formula

Registration/Randomization to Step2 - Arms A, B, C and D

* Histopathologic assessment of surgical pathology must include examination for perineural invasion (PNI) and lymphovascular invasion (LVI) and reported as absent or present; the absence or presence of extracapsular extension (ECE) requires gross and microscopic assessment and is defined to be:

  * Absent (negative or nodal metastasis with smooth/rounded leading edge confined to thickened capsule/pseudocapsule),
  * Present - minimal (tumor extends =\< 1 mm beyond the lymph node capsule), or
  * Present - extensive (gross, tumor extends \> 1 mm beyond the lymph node capsule (includes soft tissue metastasis)
* Patient must be stratified/classified into one of the following risk categories (the highest risk feature assessed pathologically will determine the patient's category/treatment arm assignment):

  * Low Risk: T1-T2, N0-N1 AND clear (≥ 3mm) margins, AND no ECE or PNI/LVI
  * High Risk: Any of the following features: one or more positive margin(s) with any T stage, OR "Extensive" (\> 1mm) ECE, OR ≥ 5 metastatic lymph nodes (regardless of primary tumor margin status)
  * Intermediate Risk: Any of the following features: one or more "close" (\< 3mm) margin(s), OR "Minimal" (≤ 1mm) ECE, OR N2a (1 or more lymph node \> 3cm in diameter), OR N2b (2-4 lymph nodes positive, any diameter ≤ 6cm), OR with perineural invasion or lymphovascular invasion.
  * Unknown Risk: Patients found to have N2C or N3 disease on final pathologic analysis are at unknown risk for recurrence, but are not candidates for deintensified adjuvant therapy in this trial. These patients will be treated on Arm C.
  * Patients not categorized into the appropriate risk category will be considered ineligible for the study
* Patient must be registered/randomized to Step 2 within a maximum of 7 weeks following surgery
* Women of childbearing potential and sexually active males are strongly advised to use an accepted and effective method of contraception

Exclusion Criteria:

Registration to Surgery (Arm S)

* Prior radiation above the clavicles
* Evidence of extensive or "matted/fixed" pathologic adenopathy on preoperative imaging
* Women must not be pregnant or breast-feeding due to the teratogenicity of chemotherapy; all females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy; a female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: has not undergone a hysterectomy or bilateral oophorectomy; or has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months)
* Any intercurrent illness likely to interfere with protocol therapy or prevent surgical resection
* Uncontrolled diabetes, uncontrolled infection despite antibiotics or uncontrolled hypertension within 30 days prior to pre-registration