Selumetinib and Akt Inhibitor MK-2206 in Treating Patients With Refractory or Advanced Gallbladder or Bile Duct Cancer That Cannot Be Removed By Surgery

Inclusion Criteria:

* Patients must have surgically unresectable histologically confirmed biliary tract adenocarcinoma (defined as gallbladder cancer, extrahepatic and intrahepatic cholangiocarcinoma; this definition excludes ampullary cancers and all tumors of mixed histology); cytological confirmation is not allowed on this study, as tissue is needed for correlative science; fresh tissue (mandatory) AND paraffin embedded tissue (positron emission tomography \[PET\]) from tumor blocks (if available) will be required from patients before enrolling on this study
* Patients will be required to undergo a biopsy prior to enrolling on the study and will be given the option to have another biopsy around 4 weeks from initiation of treatment
* Patients must have measurable disease by RECIST 1.1 criteria, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 10 mm with spiral computed tomography (CT) scan (CT scan slice thickness no greater than 5 mm); malignant lymph nodes will be considered measurable if they are \>= 15 mm in short axis
* All of the following:

  * Patients must have received only one prior line of systemic therapy for recurrent or advanced disease
  * Prior adjuvant therapy (chemotherapy +/- radiation) completed within 6 months of diagnosis of recurrence/metastases is equivalent to one line of prior therapy for metastatic disease
  * For patients who completed adjuvant therapy \> 6 months prior to diagnosis of recurrence/metastases, progression on 1 prior line of systemic therapy for metastatic disease is required
  * No prior Akt inhibitors or mitogen-activated protein kinase kinase (MEK) inhibitors allowed
  * For patients who had having prior cryotherapy, radiofrequency ablation, ethanol injection, transarterial chemoembolization (TACE) or photodynamic therapy, the following criteria must be met

    * 6 weeks must have elapsed since that therapy
    * Indicator lesion(s) is/are outside the area of prior treatment or, if the only indicator lesion is inside the prior treatment area, there must be clear evidence of disease progression associated with that lesion
    * Edges of the indicator lesion are clearly distinct on CT scanning
  * Prior radiation therapy with or without the use of a fluoropyrimidine as a radiosensitizer in the adjuvant setting will be allowed on study if \> 12 weeks have elapsed since therapy
  * Prior palliative radiation therapy will be allowed as long as \> 2 weeks have elapsed since therapy
* Life expectancy of greater than 12 weeks
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Leukocytes\>= 3,000/µL
* Absolute neutrophil count \>= 1,500/µL
* Platelets \>= 100,000/µL
* Total bilirubin =\< 1.5 X institutional upper limit of normal
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal OR =\< 5 X institutional upper limit of normal for intrahepatic cholangiocarcinoma if thought to be related to disease
* Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal
* The effects of MK-2206 and AZD6244 hydrogen sulfate on the developing human fetus at the recommended therapeutic dose are unknown; for this reason and because AZD6244 hydrogen sulfate and MK-2206 is known to be teratogenic, women of childbearing potential and men must use two forms of contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, the patient should inform the treating physician immediately
* Ability to understand and the willingness to sign a written informed consent document
* Ability to swallow oral tablets and capsules

Exclusion Criteria:

* Patients who have had chemotherapy, biologic therapy, or immunotherapy, within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events to a grade 1 or less due to agents administered more than 4 weeks earlier
* Patients may not be receiving any other investigational agents
* Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to MK-2206 or AZD6244 hydrogen sulfate or other agents used in the study
* Preclinical studies demonstrated the potential of MK-2206 for induction of hyperglycemia in all preclinical species tested; patients with diabetes or in risk for hyperglycemia should not be excluded from trials with MK-2206, but the hyperglycemia should be well controlled on oral agents before the patient enters the trial
* Preclinical studies indicated transient changes in corrected QT (QTc) interval during MK-2206 treatment; prolongation of QTc interval is potentially a safety concern while on MK-2206 therapy; cardiovascular: baseline corrected QT by Fridericia's formula (QTcF) \> 450 msec (male) or QTcF \> 470 msec (female) will exclude patients from entry on study; a list of medications that may cause QTc interval prolongation should be avoided by patients entering on trial
* Patients with clinically significant bundle branch block or pre-existing clinically significant bradycardia will be excluded from the study
* History of any other malignancy other than biliary cancer in the last 3 years, except for adequately treated basal cell carcinoma, and squamous cell carcinoma of the skin or cervix
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study; developmental and reproductive toxicity studies of MK-2206 and AZD6244 hydrogen sulfate have not been performed thus far; women of child-bearing potential and men participating in clinical studies of AZD6244 hydrogen sulfate and MK-2206 must use appropriate contraception, including abstinence and double-barrier methods, throughout AZD6244 hydrogen sulfate and MK-2206 therapy; in preclinical mutagenicity studies, ADZ6244 hydrogen sulfate and MK-2206 were neither genotoxic or mutagenic; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with AZD6244 hydrogen sulfate and MK-2206, breastfeeding should be discontinued if the mother is treated with AZD6244 hydrogen sulfate and MK 2206
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with AZD6244 hydrogen sulfate and MK-2206; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated
* Patients requiring strong inhibitors or inducers of cytochrome P450 3A4 (CYP3A4) or those receiving any medications or substances that are inhibitors or inducers of CYP 450 3A4 are ineligible