Novel SEQUEnced Immunotherapy With Anti-angiogenesis and Chemotherapy in Advanced gastroesophageaL Adenocarcinoma

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information prior to registration. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Age ≥ 18 years at the time of consent. * ECOG Performance Status (PS) of 0-1within 28 days prior to registration. NOTE: Within 0-3 days prior to the anticipated C1D1, ECOG PS must be 0-1. * Tumor tissue must be obtained from a biopsy performed either (a) prior to registration or (b) prior to C1D1, as described below. * Prior to registration: if a biopsy was performed prior to registration and no interval systemic anti-cancer treatment was administered between the biopsy collection and C1D1, part of that tissue is required for correlative analysis, and must be identified during screening and shipped after registration. In this situation, tissue from a new biopsy is not required. * Prior to C1D1: For all other subjects, a new biopsy prior to C1D1 is required (research biopsy per parallel biopsy protocol entitled: "Exploration of tumor biology in patients with metastatic esophageal and gastric cancer", \[biorepository protocol for prospective tissue collection\]) to obtain tissue. * NOTE: If tissue cannot be obtained by either of the above approaches (e.g., clinically contraindicated), the subject is not eligible for trial participation. * Willingness to provide tissue and blood samples for correlative research purposes and presence of a malignant lesion that is amenable to repeat biopsy while on study protocol (e.g., primary tumor that can be accessed by EGD). --NOTE: Enrollment in parallel biopsy protocol, if open for enrollment, is required. Parallel biopsy protocol entitled: "Exploration of tumor biology in patients with metastatic esophageal and gastric cancer (biorepository protocol for prospective tissue collection)". * PD-L1 results are required, if available. If PD-L1 testing has not been done, it should be ordered as standard of care prior to C1D1. PD-L1 testing must be performed by a CLIA certified lab using the Dako 22C3 antibody. * Histologically or cytologically proven adenocarcinoma of the stomach or GEJ. * Metastatic, recurrent, or locally advanced unresectable disease. * Candidate for pembrolizumab, ramucirumab, and paclitaxel (or nab-paclitaxel) * Demonstrate adequate organ function as defined in the table below. All screening labs to be obtained within 28 days prior to registration. NOTE: Labs must also be obtained within 10 days prior to C1D1 treatment. * Absolute Neutrophil Count (ANC) ≥ 1,100/mm3 * Hemoglobin (Hgb) ≥ 8.5 g/dL without transfusion or EPO dependency * Platelets ≥ 100,000 / mcL * Creatinine OR Calculated creatinine clearance (institutional standard for calculation of CrCl may be used) ≤ 1.5 x upper limit of normal (ULN) OR ≥ 60 mL/min for subject with creatinine levels \> 1.5 x institutional ULN * Total bilirubin ≤ 1.5 X ULN OR Direct bilirubin ≤ ULN for subjects with total bilirubin levels \> 1.5 ULN OR total bilirubin ≤ 2 x ULN if liver metastases are present (patients with Gilbert's syndrome are allowed) * Aspartate aminotransferase (AST) and Alanine aminotransferase (ALT) ≤ 2.5 X ULN OR ≤ 5 x ULN for subjects with liver metastases * Albumin \> 3.0 g/dL * Females of childbearing potential must have a negative pregnancy test within 72 hours prior to registration. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. NOTE: Females are considered of child bearing potential unless they are surgically sterile (have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are naturally postmenopausal for at least 12 consecutive months * Females of childbearing potential and males must be willing to abstain from heterosexual intercourse or to use 2 forms of effective methods of contraception from the time of informed consent (females)/prior to C1D1 (males) until 120 days after treatment discontinuation. * Willingness to return to the enrolling institution for follow up Exclusion Criteria for all cohorts: * Past medical history of interstitial lung disease (ILD), drug-induced ILD, radiation pneumonitis which required steroid treatment, or any evidence of clinically active interstitial lung disease (relevant for ramucirumab). * Any of the following cardiac criteria: * Clinically important abnormalities in rhythm, conduction or morphology of resting ECG e.g., clinically important forms of complete left bundle branch block, third degree heart block, second degree heart block, mean resting corrected QT interval (QTc using Fridericia's formula) \> 470 msec, PR interval \>250msec. NOTE: Investigators are encouraged to discuss potentially clinically important arrythmias with a cardiologist and consider implementing more frequent monitoring if appropriate (e.g., more frequent ECGs). * Symptomatic heart failure, uncontrolled hypokalemia despite repletion, congenital long QT syndrome, family history of long QT syndrome or unexplained sudden death under 40 years of age in first degree relatives. NOTE: Factors that increase risk of QTc prolongation or risk of arrhythmia, such as concomitant medications, may require increased monitoring during Combination Therapy (See Section 7). * The patient has experienced any arterial thromboembolic events, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina, within 6 months prior to first dose of protocol therapy. * The patient has uncontrolled or poorly-controlled hypertension (\>160 mmHg systolic or \> 100 mmHg diastolic for \>4 weeks) despite standard medical management. * The patient has experienced any Grade 3-4 GI bleeding within 3 months prior to first dose of protocol therapy (relevant for ramucirumab). * Cirrhosis at a level of Child-Pugh B (or worse) or cirrhosis (any degree) with a history of hepatic encephalopathy or clinically meaningful ascites resulting from cirrhosis. Clinically meaningful ascites is defined as ascites from cirrhosis requiring diuretics or paracentesis (relevant for ramucirumab). * Hemoptysis (defined as bright red blood or ≥ 1/2 teaspoon) within 2 months prior to first dose of protocol therapy or with radiographic evidence of intratumor cavitation or has radiologically documented evidence of major blood vessel invasion or encasement by cancer (relevant for ramucirumab) that in the opinion of the investigator poses unacceptably high risk as standard clinical practice when combined with ramucirumab. * The patient has a prior history of GI perforation/fistula (within 6 months of first dose of protocol therapy) that in the opinion of the investigator poses unacceptably high risk as standard clinical practice when combined with ramucirumab. * The patient has a serious or nonhealing wound, ulcer, or bone fracture within 28 days prior to first dose of protocol therapy that in the opinion of the investigator poses unacceptably high risk as standard clinical practice when combined with ramucirumab. * The patient has undergone major surgery within 28 days prior to first dose of protocol therapy prior to the first dose of protocol therapy. The patient has elective or planned major surgery to be performed during the course of the clinical trial (relevant for ramucirumab). * 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. * Has active autoimmune disease that has required systemic treatment in the past 6 months (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs) that in the opinion of the investigator poses unacceptably high risk as standard clinical practice when combined with pembrolizumab. Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. NOTE: Inhaled steroids or steroid injections for joint disease are allowed. * Is currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment. * Prior therapy combining anti-angiogenesis agent with cytotoxic agent(s). NOTE: Prior single-agent anti-angiogenesis therapy (eg, ramucirumab monotherapy) is allowed. * Patients known to be HIV positive. * Has known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA \[qualitative\] is detected). * Prior severe allergic reactions to a monoclonal antibody or hypersensitivity to pembrolizumab or any of its excipients. * Has had a prior anti-cancer monoclonal antibody (mAb), excluding anti-PD-1/-L1 therapy, within 3 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 3 weeks earlier. * Other active malignancy which requires current treatment and which in the opinion of the site investigator is likely to interfere with evaluation of disease assessment. NOTE: Continuation of hormonal therapies is allowed. * Patients with known active central nervous system (CNS) metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability. * Uncontrolled intercurrent illness which in the opinion of the investigator poses unacceptably high risk when combined with study treatment, including but not limited to the following: * Symptomatic congestive heart failure * Unstable angina pectoris * Severely impaired lung function * Known history of active TB (Bacillus Tuberculosis) * Uncontrolled diabetes as defined by fasting serum glucose \>1.5 x ULN (NOTE: Optimal glycemic control should be achieved before starting trial therapy.) * Significant underlying liver disease such as severe cirrhosis or hepatic impairment * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Has an active infection requiring systemic therapy prior to therapy initiation and that in the opinion of the investigator poses unacceptably high risk as standard clinical practice when starting study treatment. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. * Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. * Has known history of or any evidence of active, non-infectious pneumonitis that in the opinion of the investigator poses unacceptably high risk as standard of clinical practice when starting study treatment. * Currently uncontrolled hyper/hypothyroidism or hyper/hypocortism if in the opinion of the investigator they pose unacceptably high risk when combined with study treatment. * Received live vaccine or live attenuated vaccine within 30 days prior to registration. NOTE: Administration of killed vaccines is allowed. * Prior toxicity from immune checkpoint inhibitors (eg. pancreatitis, enteritis) that was symptomatic or required medical intervention ≤ 6 months prior to registration, which in the opinion of the investigator poses unacceptably high risk as standard clinical practice when starting study treatment. * Pre-existing motor or sensory neurotoxicity grade 3 or higher. * Prior solid organ or allogeneic transplant. Eligibility specific to Cohort 1: * Inclusion: Measurable disease per RECIST v1.1 * Exclusion: Disease progression according to RECIST v1.1 or irRECIST, or treatment intolerance, during prior therapy with an anti-PD-1, anti-PD-L1, or anti- PD-L2 agent. NOTE: Stable or responsive disease on anti-PD-1/L1-L2 therapy (without concurrent cytotoxic therapy) is allowed. * Exclusion: Weight loss ≥ 5% during prior anti-PD-1, anti-PD-L1, or anti- PD-L2-containing therapy (from time of initiation of such therapy to most recent dose). Eligibility specific to Cohort 2: * Inclusion: Measurable or evaluable disease per RECIST v1.1. * Inclusion: Received anti-PD-1 therapy with or without concurrent cytotoxic chemotherapy as preceding therapy. * Inclusion: PD-L1 combined positive score (CPS) ≥ 1 or tumor proportion score (TPS) ≥ 1 at any time before registration. NOTE: PD-L1 expression status in tissue samples collected at baseline before receiving any prior systemic anticancer therapy are of highest interest.