MK2206 and Paclitaxel in Treating Patients With Locally Advanced or Metastatic Solid Tumors or Metastatic Breast Cancer

Inclusion Criteria: * Patients with histologically or cytologically confirmed locally advanced or metastatic solid tumors who have received at least two lines of therapy; for the expansion phase: female patients with metastatic breast cancer who have received a maximum of three lines of therapy * Absolute neutrophil count (ANC) \>= 1,000/uL * Platelets \>= 100,000/uL * Hemoglobin (Hgb) \>= 9 g/dL * Creatinine =\< 1.5 x upper limit of normal (ULN) * Prothrombin time (PT) within institutional guideline for biopsy procedure * Total bilirubin =\< 1.5 x ULN * Alanine aminotransferase (ALT) =\< 2.5 x ULN (=\< 3 x ULN for subjects with liver involvement with cancer) * A known diabetic patient who is taking insulin or oral anti-diabetic therapy must have a hemoglobin A1C (HBA1C) =\< 8% or a fasting serum glucose =\< 110% ULN * Patient will have a tumor suitable for fine-needle aspirates (FNA) and core biopsy for research purposes (determined by the treating physician) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Measurable disease by Response Evaluation Criteria In Solid Tumors (RECIST) or evaluable disease (e.g., bone metastasis or lesions which do not fulfill RECIST criteria for metastatic disease) * Patients with central nervous system (CNS) metastasis who have completed a course of therapy (for treatment of CNS metastasis) would be eligible for the study provided they are clinically stable for 1 month prior to entry as defined as: * No evidence of new or enlarging CNS metastasis * Off steroids and anticonvulsants * Corrected QT (QTc) interval =\< 450 msec (Bazett's formula) * Negative serum pregnancy test beta-human chorionic gonadotropin (hCG) for patients of childbearing age * For the dose escalation cohorts, patients must have received front-line, cytotoxic, systemic therapy (combination or single agent, with or without the addition of targeted agents) for advanced cancer * For the expansion cohort, patients must have received no more than three lines of cytotoxic systemic therapy (combination or single agent, with or without the addition of targeted agents) for metastatic breast cancer; patients could have received paclitaxel in the adjuvant setting, but not in the metastatic setting * The last line of therapy must have been administered \> 21 days prior to initiation of treatment on this study * 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 * Both men and women and members of all races and ethnic groups are eligible for this trial Exclusion Criteria: * Patients may not be receiving any other investigational agents * Patients taking a potent cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inhibitor or inducer will be excluded; patients who have discontinued any of these medications must have a wash-out period of at least 5 days or at least 5 half-lives of the drug (whichever is longer) prior to the first dose MK-2206 * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MK-2206 or other agents used in the study * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, bradycardia, related to cardiac disease, bundle branch block, 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; breastfeeding should be discontinued