Weekly Doses of Cilengitide and Paclitaxel in Treating Patients With Advanced Solid Tumors That Cannot Be Removed by Surgery

Inclusion Criteria: * Histologic proof of cancer that is now unresectable (solid tumors, excluding lymphoma) * For Cohort II only: * Histologic adenocarcinoma of the breast with manifestations of metastatic cancer or locally advanced, unresectable cancer * Estrogen, progesterone, and human epidermal growth factor receptor 2 (HER2) receptor negative disease per local standards * Refractory to taxanes which is defined as one of the following: * Having relapsed during or within 12 months of completing adjuvant paclitaxel or docetaxel * Disease progression while on any taxane in the locally advanced, unresectable or metastatic breast cancer setting * Ability and willingness to undergo biopsy for biomarker testing prior to start of treatment * Disease must be measurable by imaging-based evaluation per Response Evaluation Criteria in Solid Tumors (RECIST) criteria (v1.1) * Up to 5 prior regimens of chemotherapy for metastatic disease are allowed * Absolute neutrophil count (ANC) \>= 1500/μL * Hemoglobin (Hgb) \>= 9 g/dL * Platelets (PLT) \>= 100,000/μL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) =\< 3 x ULN or AST =\< 5 x ULN if liver involvement * Creatinine =\< 1.5 x ULN * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, or 1 (or Karnofsky performance status \[KPS\] \> 70) * Ability to provide informed consent * Willingness to return to the enrolling Mayo Clinic institution for follow-up * Life expectancy \>= 12 weeks * All patients: Willingness to provide blood samples for the mandatory correlative research component * For Cohort II, tissue biopsies are mandatory * Women of childbearing potential only: negative serum pregnancy test done =\< 7 days prior to registration, for women of childbearing potential including women within 2 years of post-menopause Exclusion Criteria: * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy * 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 * Any of the following prior therapies: * Chemotherapy =\< 21 days prior to registration * Mitomycin C/nitrosoureas =\< 42 days prior to registration * Immunotherapy =\< 14 days prior to registration * Biologic therapy =\< 14 days prior to registration * Prior investigational therapy =\< 28 days prior to registration * Full field radiation therapy =\< 28 days prior to registration or limited field radiation therapy \< 14 days prior to registration * Full field radiation encompasses the entire area of known disease involvement and surrounding uninvolved but at-risk areas, e.g. subtotal nodal (mantle and upper abdomen) or total nodal irradiation * Limited field radiation is restricted to treating only the known areas of clinical disease, e.g. involved-field therapy for lymphoma * Major surgery (i.e., laparotomy) =\< 4 weeks prior to registration; minor surgery =\< 2 weeks prior to registration; Note: insertion of a vascular access device is not considered major or minor surgery in this regard * Unresolved toxicities from prior therapy with the exception of alopecia that have not resolved to =\< grade 1, unless the patient has a chronic, stable =\< grade 2 toxicity that would not interfere with the evaluation of the study agents * New York Heart Association classification III or IV * Central nervous system (CNS) metastases or seizure disorder; Note: CNS metastases are allowed if previously treated and stable for at least 4 weeks * Any of the following: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception (condoms, diaphragm, injections, intrauterine device \[IUD\], or abstinence, etc.); oral, implantable, or injectable contraceptives may be affected by cytochrome P450 interactions, and are therefore not considered effective for this study * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration \[FDA\]-approved indication and in the context of a research investigation) * 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 safety and toxicity of the prescribed regimens * Immunocompromised patients (other than that related to the use of corticosteroids) including patients known to be human immunodeficiency virus (HIV) positive on highly active antiretroviral therapy (HAART) therapy * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy =\< 3 years prior to registration; EXCEPTIONS: nonmelanotic skin cancer or carcinoma-in-situ of the cervix; Note: if there is a history of prior malignancy, they must not be receiving other specific treatment for their cancer * History of myocardial infarction =\< 6 months prior to registration, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Uncontrolled hypertension, labile hypertension of history of poor compliance with antihypertensive medication * Patients with active, bleeding diathesis * Non-disease related- major surgery, =\< 28 days or minor surgery =\< 7 days prior to registration