Multicohort Phase II Trial of sEphB4-HSA+Pembrolizumab in Solid Tumors

Inclusion Criteria (for all Cohorts):

* Be willing and able to provide written informed consent/assent for the trial
* Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) performance scale
* Absolute neutrophil count (ANC) \>= 1,500/mcL
* Platelets \>= 100,000/mcL
* Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within 7 days of assessment)
* Measured or calculated creatinine clearance (glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) \>= 30mL/min for subject with creatinine levels \> 1.5 X institutional upper limit of normal (ULN)
* Serum total bilirubin =\< 1.5 X ULN or direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN or =\< 5 X ULN for subjects with liver metastases
* Albumin \>= 2.5 mg/dL
* International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants
* Activated partial thromboplastin time (aPTT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants
* Recovered to grade 1 from the effects (excluding alopecia) of any prior therapy for their malignancies
* Female subject of childbearing potential should have a negative urine or serum pregnancy; if the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required
* Female subjects of childbearing potential must be willing to use adequate method of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication; subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year
* Male subjects must agree to use an adequate method of contraception starting with the first dose of study therapy through 120 days after the last dose of study therapy

Criteria specific to 2nd line and 3rd line and beyond cohorts (Cohorts A and B):

* Advanced (metastatic or recurrent) pathologically proven urothelial carcinoma which is refractory to platinum based due to disease progression on a platinum containing regimen; patients progressing within 12 months of their last dose of platinum-based neoadjuvant or adjuvant chemotherapy will be considered platinum refractory
* Have measurable disease based on RECIST 1.1
* Be willing to provide tissue from a newly obtained core or excisional biopsy of a tumor lesion; newly-obtained is defined as a specimen obtained for up to 12 weeks (84 days) after discontinuation of previous systemic therapy and prior to initiation of treatment on day 1 on this study; subjects for whom newly-obtained samples cannot be provided (e.g. inaccessible or subject safety concern) may submit an archived specimen only upon agreement from the Sponsor; an optional core biopsy will be requested from an accessible metastatic site after 2 cycles of treatment and prior to progression of disease to help the investigators better understand the activity of these drugs in tumor tissue.

Criteria specific to the neoadjuvant urothelial cohort (Cohort C):

* Must have tumor stage ≥T2 AND ≤T4a, N0, M0 (AJCC 8th edition) urothelial carcinoma of the bladder deemed resectable and planned for radical cystectomy with curative intent. T4 due to infiltration of the prostate is allowed.
* Must have TURBT specimen obtained within 12 weeks prior to the first day of treatment on the study and the specimen must include muscle. Subjects for whom a TURBT specimen within 12 weeks of C1D1 cannot be provided, but an older specimen is available (e.g. unlikely to be able to obtain adequate specimen or subject safety concern) may submit an archived specimen only upon written agreement from the Sponsor.
* Prior intravesical therapy is allowed. However, patients who have received prior systemic therapy within 12 months enrollment are excluded.

Criteria specific to the neoadjuvant prostate cohort (Cohort D):

* Must have biopsy proven prostate cancer (Gleason Score ≥7, and PSA \>4.0 ng/mL- rare cases can be reviewed and approved with a written agreement from the Sponsor) amenable to radical prostatectomy.
* Must have appropriate staging imaging showing no evidence of distant metastatic disease. Choice of imaging is per treating physician- some acceptable imaging examples include MRI of pelvis, CT of abdomen and pelvis, bone scan, Axumin PET CT, and PSMA PET CT. For Cohort D, the imaging studies may be considered valid for enrollment beyond the 28 days at the discretion of the treating physician and no longer than 90 days.

Exclusion Criteria:

* 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
* Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment
* Has a known history of active TB (bacillus tuberculosis)
* Hypersensitivity to pembrolizumab or any of its excipients
* Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study day 1 or who has not recovered (ie, =\< grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier
* Has had prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study day 1 or who has not recovered (ie, =\< grade 1 or at baseline) from adverse events due to a previously administered agent; Note: subjects with =\< grade 2 neuropathy are an exception to this criterion and may qualify for the study; Note: if subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy
* Has a known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer.
* Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis; subjects with previously treated brain 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
* Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs); replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc) is not considered a form of systemic treatment
* Has known history of, or any evidence of active, non-infectious pneumonitis
* Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial
* 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 received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent, or sEsphB4-HSA
* Has a known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies)
* Has known active Hepatitis B (eg, hepatitis B surface antigen \[HBsAg\] reactive) or Hepatitis C (eg, hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected)
* Has New York Heart Association (NYHA) class 3 or 4, myocardial infarction, acute coronary syndrome, diabetes mellitus with ketoacidosis or chronic obstructive pulmonary disease (COPD) requiring hospitalization in the preceding 6 months; or any other intercurrent medical condition that contraindicates treatment with sEphB4HSA or pembrolizumab (MK-3475) or places the patient at undue risk for treatment related complications
* Has received a live vaccine within 30 days of planned start of study therapy; Note: seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (eg, flu-mist) are live attenuated vaccines, and are not allowed
* Uncontrolled hypertension is excluded- systolic blood pressure \>140mmHg or diastolic \>90mmHg. Patients experiencing white coat hypertension in the office, may be considered eligible if blood pressure log at home is within acceptable limits AND upon review and agreement from the Sponsor.