Lenvatinib and Everolimus in Treating Patients With Advanced, Unresectable Carcinoid Tumors

Inclusion Criteria: * Histologically or cytologically confirmed unresectable well differentiated (irrespective of grade) carcinoid tumors. Patients with multiple neuroendocrine tumors associated with MEN1 syndrome will be eligible * Patients must have radiographically measurable disease. Lesions that have had locoregional therapies such as radiofrequency (RF) ablation, radiation or transarterial therapies must show evidence of progressive disease based on RECIST 1.1 to be deemed a target lesion. * Patients with other gastrointestinal neuroendocrine tumors must have had progressive disease over the last 12 months irrespective of number of prior therapies. Patients with both functional (who may continue somatostatin analogues as required for control of related symptoms) and non-functional tumors are eligible. * Written informed consent must be obtained prior to any screening procedures. * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 - 1. * A sufficient interval must have elapsed between the last dose of prior anti-cancer therapy (including cytotoxic and biological therapies and major surgery) and enrollment, to allow the effects of prior therapy to have abated: * Cytotoxic or targeted chemotherapy: \>= the duration of the cycle of the most recent treatment regimen (a minimum of 3 weeks for all regimens, except 6 weeks for nitrosoureas and mitomycin-C) * Biologic therapy (e.g., antibodies): \>= 4 weeks. * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L. * Hemoglobin (Hgb) \>= 9 g/dL. * Platelets \>= 100 x 10\^9/L. * Serum total bilirubin =\< 1.5 x upper limit of normal (ULN). * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x ULN, except in patients with tumor involvement of the liver who must have AST and ALT =\< 5 x ULN. * Serum creatinine =\< 1.5 x ULN or 24-hour clearance \>= 50 mL/min. * Serum potassium, sodium, magnesium, phosphorus and total calcium (corrected for serum albumin) must be within clinically relevant limits (e.g., a patient can be enrolled if a lab value may be outside the normal laboratory range but the abnormality is not clinically relevant or can be repeated.). * Spirometry and diffusion capacity of the lung for carbon monoxide (DLCO) 50% or greater of normal and oxygen (O2) saturation 88% or greater at rest on room air. Baseline testing is not required unless known severely impaired lung function. * Negative pregnancy test (serum beta-human chorionic gonadotropin \[HCG\]) within 7 days of starting study treatment is required in women of childbearing potential. Beta-HCG may be secreted by a small percentage of neuroendocrine tumors (NETs) and be a tumor marker. Thus, NET patients with positive beta-HCG are eligible if pregnancy can be excluded by vaginal ultrasound or lack of expected doubling of beta-HCG. Exclusion Criteria: * Patient has a known hypersensitivity to lenvatinib, everolimus or any of their excipients. * Patients with known or suspected brain metastases. However, if radiation therapy and/or surgery has been completed and serial evaluation by computed tomography (CT) (with contrast enhancement) or magnetic resonance imaging (MRI) over a minimum of 3 months demonstrates the disease to be stable and if the patient remains asymptomatic, then the patient may be enrolled. Such patients must have no need for treatment with steroids or anti-epileptic medications. * Patients with concurrent malignancies or malignancies within 3 years prior to starting study drug (with the exception of tumors common to a single genetic cancer syndrome, i.e. MEN1, MEN2, vHL, TSC etc., or adequately treated, basal cell skin cancer, squamous cell carcinoma, non-melanoma skin cancer or curatively resected cervical cancer). * Patient is not able to swallow oral medication and/or has impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of the study drugs (e.g., ulcerative diseases, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, or small bowel resection). * Known diagnosis of human immunodeficiency virus (HIV), hepatitis B or hepatitis C (testing is not mandatory). * Patient who has received radiotherapy within =\< 4 weeks or limited field radiation for palliation within =\< 2 weeks prior to starting study drug, and who has not recovered to grade 1 or better from related side effects of such therapy (exceptions include alopecia) and/or in whom \>= 30% of the bone marrow was irradiated. * Patient has had major surgery within 21 days prior to starting study drug or has not recovered from major side effects (tumor biopsy is not considered as major surgery). * Impaired cardiac function or clinically significant cardiac diseases, including any of the following: * History of acute coronary syndromes (including myocardial infarction, unstable angina, coronary artery bypass grafting, coronary angioplasty, or stenting) or symptomatic pericarditis \< 12 months prior to screening * History of documented congestive heart failure (New York Heart Association functional classification III-IV) * Documented cardiomyopathy * Patient has a left ventricular ejection fraction (LVEF) \< 50% as determined by multiple gated acquisition (MUGA) scan or echocardiogram (ECHO) at screening * History of ventricular, supraventricular, nodal arrhythmias, or any other cardiac arrhythmias, long QT Syndrome or conduction abnormality within 12 months prior to starting study drug * Congenital long QT syndrome or a family history of QTc prolongation * On screening, inability to determine the corrected QT interval using Fridericia's formula (QTcF) interval on the electrocardiogram (ECG) (i.e.: unreadable or not interpretable) or QTcF \> 450 msec (using Fridericia's correction). * Systolic blood pressure \> 150 mmHg or diastolic blood pressure \> 90 mmHg. Patients with systolic blood pressure \< 150 mmHg or diastolic blood pressure \< 90 mmHg on anti-hypertensives without any change in anti-hypertensives within 1 week prior to screening visit are eligible. * Patients with concurrent severe and/or uncontrolled concurrent medical conditions that could compromise participation in the study (e.g., uncontrolled diabetes mellitus defined by a glucose \> 1.5 ULN in spite of adequate medical treatment, clinically significant pulmonary disease, clinically significant neurological disorder, active or uncontrolled infection, fasting total cholesterol \> 300 mg/dL (or \> 7.75 mmol/L) or fasting triglycerides level \> 2.5 x ULN in spite of optimal medical management * Patient has a history of non-compliance to medical regimen or inability to grant consent. * Pregnant or lactating women, where pregnancy is defined as the state of a female after conception and until the termination of gestation, confirmed by a positive hCG laboratory test (\> 5 mIU/mL). * Women of child-bearing potential, defined as all women physiologically capable of becoming pregnant, unless they are using highly effective methods of contraception throughout the study and for 8 weeks after study drug discontinuation. Highly effective contraception methods include: * Total abstinence when this is in line with the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, post ovulation methods) and withdrawal are not acceptable methods of contraception. * Female sterilization (have had surgical bilateral oophorectomy with or without in case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment. * Male sterilization (at least 6 months prior to screening). For female patients on the study, the vasectomized male partner should be the sole partner for that patient. * Combination of any of the two following * Use of oral, injected or implanted hormonal methods of contraception or other forms of hormonal contraception that have comparable efficacy (failure rate \< 1%), for example hormone vaginal ring or transdermal hormone contraception * Placement of an intrauterine device (IUD) or intrauterine system (IUS) * Barrier methods of contraception: condom or occlusive cap (diaphragm or cervical/vault caps) with spermicidal foam/gel/film/cream/ vaginal suppository. * In case of use of oral contraception, women should have been stable on the same pill before taking study treatment. Note: Oral contraceptives are allowed but should be used in conjunction with a barrier method of contraception due to unknown effect of drug-drug interaction. * Women are considered post-menopausal and not of child bearing potential if they have had 12 months of natural (spontaneous) amenorrhea with an appropriate clinical profile (e.g. age appropriate, history of vasomotor symptoms) or have had surgical bilateral oophorectomy (with or without hysterectomy) or tubal ligation at least six weeks ago. In the case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment is she considered not of child bearing potential. * Sexually active males unless they use a condom during intercourse while taking the drug and for 28 days after stopping treatment and should not father a child in this period. A condom is required to be used also by vasectomized men in order to prevent delivery of the drug via seminal fluid. * Patients unwilling or unable to comply with the protocol. * Patient is currently receiving warfarin or other coumarin-derived anticoagulant for treatment, prophylaxis or otherwise. Therapy with heparin, low molecular weight heparin (LMWH) or fondaparinux is allowed. * Subjects having \> 1 + proteinuria on urine dipstick testing will undergo 24 hour (h) urine collection for quantitative assessment of proteinuria. Subjects with urine protein \>= 1 g/24 h will be ineligible. * Active hemoptysis (bright red blood of at least 0.5 teaspoon) within 3 weeks prior to the first dose of study drug. * Prior therapy with mTOR inhibitors (e.g. sirolimus, temsirolimus, deforolimus) and/or lenvatinib. Prior antiangiogenic therapies (including but not limited to bevacizumab, aflibercept, sunitinib and/or pazopanib) are allowed. * Other active malignancy (except definitively treated melanoma in-situ, basal or squamous cell carcinoma of the skin, or carcinoma in-situ of the cervix or bladder) within past 24 months.