Akt Inhibitor MK2206, Lapatinib Ditosylate, and Trastuzumab in Treating Patients With Locally Advanced or Metastatic HER2-Positive Breast , Gastric, or Gastroesophageal Cancer That Cannot Be Removed By Surgery

Inclusion Criteria:

* Patients must have histologically confirmed HER2-positive invasive breast, gastric, or gastroesophageal carcinoma that is locally advanced and unresectable or metastatic and for which standard curative or palliative measures do not exist or are no longer effective; HER2-positive is defined as HER2 overexpression and/or amplification as determined by immunohistochemistry (3+) or fluorescence in situ hybridization (FISH) (\>= 2.0)
* Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan
* Patients may have previously received trastuzumab or lapatinib as part of a regimen in the adjuvant or metastatic setting with evidence of progression
* No restriction on prior chemotherapy regimens for advanced stage disease; no restriction for prior hormonal therapy
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Life expectancy of greater than three months
* Left ventricular ejection fraction (LVEF) by multi-gated radionuclide angiography scan (MUGA) or echocardiogram (ECHO) at or above the lower limit of normal of 50%
* Leukocytes \>= 3,000/mcL
* Absolute neutrophil count \>= 1,000/mcL
* Platelets \>= 100,000/mcL
* Total bilirubin within normal institutional limits
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal
* 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 on the developing human fetus are unknown; for this reason, women of child-bearing potential and men must agree to use adequate 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, she should inform her treating physician immediately
* Patients must be able to swallow whole tablets; nasogastric or gastrostomy (G) tube administration is not allowed; tablets must not be crushed or chewed
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier; if the patient has residual toxicity from prior treatment, toxicity must be =\< grade 1
* Patients who are receiving any other investigational agents; no concurrent use of endocrine therapy is permitted; patients on bisphosphonates or denosumab for bone metastases or osteopenia/porosis are considered eligible
* Patients with active brain metastases requiring radiation should be excluded as they may develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; patients with stable brain metastases (mets) (\> 6 months without change in steroids or seizure medications) will be considered eligible
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to MK-2206, trastuzumab, or lapatinib
* Patients receiving any medications or substances that are inhibitors or inducers of cytochrome P450 3A4 (CYP450 3A4) are ineligible
* 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 or insulin before the patient enters the trial; patients with poorly controlled diabetes with hemoglobin (Hgb)A1C \> 9% will be excluded
* Preclinical studies indicated transient changes in corrected QT interval (QTc) interval during MK-2206 treatment; prolongation of QTc interval is potentially a safety concern while on MK-2206 therapy; cardiovascular baseline QTc \> 480 msec will exclude patients from entry on study
* 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 because MK-2206 an agent with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with MK-2206, breastfeeding should be discontinued if the mother is treated with MK-2206; these potential risks may also apply to other agents used in this study
* Patients with unstable angina, congestive heart failure, or with a history of a myocardial infarction within 6 months; patients with high-risk uncontrolled arrhythmias (ventricular tachycardia, high-grade atrioventricular \[AV\] block, supraventricular arrhythmias which are not adequately rate-controlled); patients with uncontrolled hypertension (i.e., over 160/90 mmHg); patients who are controlled on anti-hypertensive medication will be allowed to enter the study
* Patients known to be human immunodeficiency virus (HIV)-positive and are on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with MK-2206; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy and as such patients with cluster of differentiation (CD)4 counts \< 200 will be excluded; HIV-positive patients not on anti-retrovirals and with an adequate CD4 count will be considered eligible