Inclusion Criteria:

* Patients in Group F must have a body surface area (BSA) \>= 0.6 m\^2
* Patients must have a performance status corresponding to Eastern Cooperative Oncology Group (ECOG) scores of 0, 1, or 2; use Karnofsky for patients \> 16 years of age and Lansky for patients =\< 16 years of age; patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score
* Patients must be newly diagnosed with histologically-proven primary pediatric hepatic malignancies including hepatoblastoma or hepatocellular carcinoma, except as noted below; patients with a diagnosis of hepatocellular neoplasm, not otherwise specified, should be classified and treated per hepatoblastoma treatment arms; note that rapid central pathology review is required in some cases; please note: all patients with histology as assessed by the institutional pathologist consistent with pure small cell undifferentiated (SCU) HB will be required to have testing for INI1/SMARCB1 by immunohistochemistry (IHC) according to the practices at the institution
* Patients with histology consistent with pure SCU must have positive INI1/SMARCB1 staining
* For all Group A patients, WDF status as determined by rapid review will be used to further stratify patients to Group A1 or A2

* For Groups B, C and D, rapid review is required if patients are either \>= 8 years of age or have an alphafetoprotein (AFP) =\< 100 at diagnosis
* For all Groups E and F patients, rapid central pathology review is required
* In emergency situations when a patient meets all other eligibility criteria and has had baseline required observations, but is too ill to undergo a biopsy safely, the patient may be enrolled without a biopsy

* Clinical situations in which emergent treatment may be indicated include, but are not limited to, the following circumstances:

* Anatomic or mechanical compromise of critical organ function by tumor (e.g., respiratory distress/failure, abdominal compartment syndrome, urinary obstruction, etc.)
* Uncorrectable coagulopathy
* For a patient to maintain eligibility for AHEP1531 when emergent treatment is given, the following must occur:

* The patient must have a clinical diagnosis of hepatoblastoma, including an elevated alphafetoprotein (AFP), and must meet all AHEP1531 eligibility criteria at the time of emergent treatment
* Patient must be enrolled on AHEP1531 prior to initiating protocol therapy; a patient will be ineligible if any chemotherapy is administered prior to AHEP1531 enrollment
* Note: If the patient receives AHEP1531 chemotherapy emergently PRIOR to undergoing a diagnostic biopsy, pathologic review of material obtained in the future during either biopsy or surgical resection must either confirm the diagnosis of hepatoblastoma or not reveal another pathological diagnosis to be included in the analysis of the study aims
* Patients may have had surgical resection of the hepatic malignancy prior to enrollment; all other anti-cancer therapy for the current liver lesion is prohibited
* Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 or

* A serum creatinine based on age/gender as follows:

* Age: maximum serum creatinine (mg/dL)
* 1 month to \< 6 months: 0.4 (male and female)
* 6 months to \< 1 year: 0.5 (male and female)
* 1 to \< 2 years: 06 (male and female)
* 2 to \< 6 years: 0.8 (male and female)
* 6 to \< 10 years: 1 (male and female)
* 10 to \< 13 years: 1.2 (male and female)
* 13 to \< 16 years: 1.5 (male), 1.4 (female)
* \>= 16 years: 1.7 (male), 1.4 (female)
* Total bilirubin =\< 5 x upper limit of normal (ULN) for age
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) \< 10 x upper limit of normal (ULN) for age
* Shortening fraction of \>= 28% by echocardiogram (for patients on doxorubicin-containing regimens \[Groups C, D, E2, and F\] assessed within 8 weeks prior to study enrollment) or
* Ejection fraction of \>= 47% by echocardiogram or radionuclide angiogram (for patients on doxorubicin-containing regimens \[Groups C, D, E2, and F\] assessed within 8 weeks prior to study enrollment)
* Group F patients only: QT/corrected QT (QTc) interval =\< 450 milliseconds for males and =\< 470 milliseconds for females (assessed within 8 weeks prior to study enrollment)
* Normal pulmonary function tests (including diffusion capacity of the lung for carbon monoxide \[DLCO\]) if there is clinical indication for determination (e.g. dyspnea at rest, known requirement for supplemental oxygen) (for patients receiving chemotherapy \[Groups A, B, C, D, E2, F\]); for patients who do not have respiratory symptoms or requirement for supplemental oxygen, pulmonary function tests (PFTs) are NOT required
* All patients and/or their parents or legal guardians must sign a written informed consent
* All institutional, Food and Drug Administration (FDA), and National Cancer Institute (NCI) requirements for human studies must be met

Exclusion Criteria:

* Prior chemotherapy or tumor directed therapy (i.e. radiation therapy, biologic agents, local therapy (embolization, radiofrequency ablation, and laser); therefore, patients with a pre-disposition syndrome who have a prior malignancy are not eligible
* Patients who are currently receiving another investigational drug
* Patients who are currently receiving other anticancer agents
* Patients with uncontrolled infection
* Patients who previously received a solid organ transplant, other than those who previously received an orthotopic liver transplantation (OLT) as primary treatment of their hepatocellular carcinoma
* Patients with hypersensitivity to any drugs on their expected treatment arm
* Group C: Patients who have known deficiency of dihydropyrimidine dehydrogenase (DPD)
* Group D:

* Patients with chronic inflammatory bowel disease and/or bowel obstruction
* Patients with concomitant use of St. John's wort, which cannot be stopped prior to the start of trial treatment
* Group F:

* Patients with peripheral sensitive neuropathy with functional impairment
* Patients with a personal or family history of congenital long QT syndrome
* These criteria apply ONLY to patients who may receive chemotherapy (all groups other than Group E1):

* Female patients who are pregnant since fetal toxicities and teratogenic effects have been noted for several of the study drugs; a pregnancy test is required for female patients of childbearing potential
* Lactating females who plan to breastfeed their infants
* Sexually active patients of reproductive potential who have not agreed to use an effective contraceptive method for the duration of their study participation

* Note for Group F: patients of childbearing potential should use effective birth control during treatment with sorafenib and for at least 2 weeks after stopping treatment