Capecitabine and Lapatinib Ditosylate With or Without Cixutumumab in Treating Patients With Previously Treated HER2-Positive Stage IIIB-IV Breast Cancer

Inclusion Criteria: * Histologically confirmed, locally advanced: (a T4 primary tumor and stage IIIB or IIIC disease) or metastatic breast cancer that has progressed after treatment with regimens that included trastuzumab and either an anthracycline or a taxane or both * NOTE: Agents need not have been given concurrently, nor in the same regimen * NOTE: Prior treatment with trastuzumab is required unless there is a contraindication for trastuzumab treatment * Pre-treatment requirements: * Prior treatment with trastuzumab in the neo-adjuvant, adjuvant or metastatic setting is required * NOTE: Prior treatment with trastuzumab is required unless there is a contraindication for trastuzumab treatment * NOTE: Concomitant use of trastuzumab is not allowed in this study * Prior chemotherapy allowed in the neo-adjuvant, adjuvant, or metastatic setting; unlimited prior chemotherapy is allowed * Prior hormonal therapy allowed in the neo-adjuvant, adjuvant, or metastatic setting; unlimited prior hormonal therapy is allowed * HER2 positive, defined as: * Validated immunohistochemistry (IHC) assay score of 3+ (defined as uniform, intense staining of \> 30% of invasive tumor cells) * -OR- Average HER2 gene copy number of \> 6 * -OR- Gene amplified (HER2:D17Z1 ratio \> 2.20) * Must have measurable disease according to Response Evaluation Criteria in Solid Tumors (RECIST) criteria * Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Hemoglobin \> 9.0 g/dL (obtained =\< 7 days prior to registration) * White blood cells (WBC) \>= 3,000/mL (obtained =\< 7 days prior to registration) * Absolute neutrophil count (ANC) \>= 1500/mL (obtained =\< 7 days prior to registration) * Platelet count \>= 75,000/mL (obtained =\< 7 days prior to registration) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (obtained =\< 7 days prior to registration) * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) and serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 2.5 x ULN or SGOT (AST) and SGPT (ALT) =\< 5 x ULN if elevations are due to liver metastases (obtained =\< 7 days prior to registration) * Serum creatinine =\< 1.5 x ULN (obtained =\< 7 days prior to registration) * Creatinine clearance \>= 30 mL/min (calculated according to Cockcroft and Gault) (obtained =\< 7 days prior to registration) * NOTE: In patients with moderate renal impairment (calculated creatinine clearance 30-50 mL/min) at baseline, a dose reduction of the capecitabine starting dose is required * Fasting glucose \< 120 mg/dL (obtained =\< 7 days prior to registration) * NOTE: Patients with diabetes are allowed to participate, provided that their blood glucose is within the guidelines above upon enrollment * International normalization ratio (INR) =\< 1.5 x ULN (obtained =\< 7 days prior to registration) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0, 1 or 2 * Adequate cardiac function defined as an ejection fraction \>= 50% as determined by multi gated acquisition scan (MUGA) or echocardiogram * Life expectancy \> 3 months * Has written informed consent been obtained? * Willingness to return to a North Central Cancer Treatment Group (NCCTG) or other participating cooperative group institution for treatment and follow-up * Patient willing to provide tissue and blood samples for research purposes * Availability of diagnostic material (i.e., diagnostic slides confirming locally advanced/metastatic disease and HER2 stained slides) and operative and pathology reports from diagnosis of locally advanced or metastatic breast cancer * NOTE: Biopsy of recurrent disease and submission of these materials is not required if materials available from initial diagnosis of locally advanced/metastatic disease * Ability to complete questionnaire(s) by themselves or with assistance Exclusion Criteria: * Any of the following because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception (as determined by the treating physician) * Stage III or IV invasive cancer, other than breast cancer, in =\< 5 years prior to registration * Actively being treated for other malignancy, excepting non-melanotic skin cancer or carcinoma-in-situ of the cervix; if there is a history of prior malignancy, patient must not be receiving other specific treatment for their cancer * New York Heart Association class III or IV cardiovascular disease * Current, active hepatic or biliary disease except Gilbert's syndrome or asymptomatic gallstones * Evidence of active brain metastasis including leptomeningeal involvement; central nervous system (CNS) metastasis controlled by prior surgery and/or radiotherapy is allowed * NOTE: To be considered controlled, there must be at least 2 months of no symptoms or evidence of progression prior to study entry and corticosteroid therapy must have been discontinued * Major surgery, chemotherapy, or immunologic therapy =\< 4 weeks prior to registration * Radiotherapy =\< 4 weeks prior to registration, except if to a non-target lesion only; prior radiation to a target lesion is permitted only if there has been clear progression of the lesion since radiation was completed; if patient receives single dose radiation for palliation or radiation to non-target lesion, they may immediately proceed to registration without waiting; acute adverse events from radiation must have resolved to =\< Common Terminology Criteria for Adverse Events (CTCAE) version (v) 3.0 grade 1 * Prior treatment with any therapy targeting IGF-I, IGF-II or its receptors (either monoclonal antibody or tyrosine kinase inhibitor), including but not limited to any of the following (which would have been received on a previous clinical trial): * IMC-A12 (cixutumumab) * CP-751,871 (figitumumab) * AMG-479 * INSM-18 * MK0646 (h7C10) * SCH717454 (19D12, robatumumab) * R1507 * OSI-906 * BMS-754807 * PPP * NVP-AEW541 * AVE-1642 * MEDI-573 * Prior therapy with any therapy targeting HER1 (EGFR) and/or HER2 (either monoclonal antibody or tyrosine kinase) other than trastuzumab, including but not limited to any of the following: * Lapatinib (Tykerb) * Gefitinib (Iressa) * Erlotinib (Tarceva) * Cetuximab (Erbitux) * Panitumumab (Vectibix) * Currently receiving treatment with any agents that are contraindicated by study therapies: * IMC-A12 - none identified to date * Lapatinib - CYP3A4 inhibitors and inducers, including grapefruit and grapefruit juice * Capecitabine - warfarin (Coumadin), cimetidine (Tagamet), allopurinol (Lopurin), sorivudine (Usevir) or brivudine (Brivex), ketoconazole (Nizoral), itraconazole (Sporanox), ritonavir (Norvir), amprenavir (Agenerase) or indinavir (Crixivan) * Uncontrolled intercurrent illness including, but not limited to: * Poorly controlled diabetes * Ongoing or active infection * Symptomatic congestive heart failure * Unstable angina pectoris * Cardiac arrhythmia * Psychiatric illness/social situations that would limit compliance with study requirements * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of the safety of the prescribed regimens * Currently receiving treatment in a different clinical study in which investigational procedures are performed or investigational therapies are administered * Immunocompromised patients (other than that related to the use of corticosteroids) including patients known to be human immunodeficiency virus (HIV) positive with an acquired immune deficiency syndrome (AIDS)-defining illness; HIV positive patients with cluster of differentiation (CD) 4 count within institutional normal range and no history of an AIDS-defining illness are eligible; however, some antiviral/antiretroviral medications which have CYP3A4 interactions are prohibited on this study