Testing the Addition of an Anti-cancer Drug, BAY 1895344, to the Usual Chemotherapy Treatment (Cisplatin, or Cisplatin and Gemcitabine) for Advanced Solid Tumors With Emphasis on Urothelial Cancer

Inclusion Criteria: * Histologically-confirmed advanced solid tumor with measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) version (v) 1.1 criteria, for which cisplatin-based therapy would be considered appropriate, including: * Non-small cell lung cancer (NSCLC) * UC * Penile cancer * Malignant pleural mesothelioma * Small cell lung cancer * Biliary tract cancer * Esophageal and gastric cancers * Ovarian cancer * Endometrial cancer * Cervical cancer * Head and neck cancer * Triple-negative breast cancer (Her2/neu-negative, estrogen receptor \[ER\]/progesterone receptor \[PR\]-negative breast cancer) * For the expansion cohort of the triplet combination at MTD/RP2D only: * Patients with histologically confirmed advanced or unresectable urothelial carcinoma are eligible * The histology should be predominantly urothelial (\>= 50% of sample evaluated contains urothelial histology) * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of BAY 1895344 in combination with gemcitabine and cisplatin in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Availability of archival FFPE tissue * Prior cisplatin exposure of \< 300 mg/m\^2. Patients with prior cisplatin treatment must have received last cisplatin treatment \> 6 months prior to enrollment * Prior treatment with PARP inhibitors is permitted (such as olaparib, rucaparib, or other experimental inhibitors of PARP administered in a clinical trial) * Prior immune checkpoint inhibitor therapy is permitted (including anti-programmed cell death protein 1 \[PD-1\], anti-PD-ligand \[L\]1 therapy, such as pembrolizumab, nivolumab, avelumab, durvalumab, atezolizumab, or anti-cytotoxic t-lymphocyte protein 4 \[CTLA4\] therapy such as ipilimumab, or other experimental immune checkpoint pathway inhibitors administered in a clinical trial) * Leukocytes \>= 3,000/mcL * Hemoglobin \>= 9 g/dL * Neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 2 mg/dL * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 3 x institutional upper limit of normal (ULN) * Creatinine clearance \>= 40 mL/min OR glomerular filtration rate (GFR) \>= 40 mL/min/1.73 m\^2 unless data exists supporting safe use at lower kidney function values, no lower than 30 mL/min/1.73 m\^2 * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression * Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy; patients with stable brain metastases that are asymptomatic and on a stable dose of steroids are also considered eligible * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional classification. To be eligible for this trial, patients should be class 2B or better * The effects of BAY 1895344, cisplatin, and gemcitabine on the developing human fetus are unknown. For this reason and because deoxyribonucleic acid (DNA)-damage response inhibitors agents as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation and for 6 months after completion of BAY 1895344 administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 6 months after completion of BAY 1895344 administration * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Life expectancy \< 6 weeks by investigator assessment * Other active malignancy requiring treatment, except for cutaneous malignancies that require resection such as squamous cell carcinoma, basal cell carcinoma, or cutaneous melanoma, and except for prostate cancer if only on androgen deprivation therapy * Significant peripheral neuropathy (grade 2 or higher by Common Terminology Criteria for Adverse Events \[CTCAE\]) * Sensorineural hearing loss (grade 2 or higher by CTCAE) * Must NOT have had prior treatment with ATR inhibitor (prior BAY1895344 or other investigational ATR inhibitors), or current treatment with any other investigational agents * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. Patients who have targeted therapies (such as PARP inhibitors) within 2 weeks prior to entering the study * History of allergic reactions attributed to compounds of similar chemical or biologic composition to BAY 1895344 or other agents used in study * Patients receiving any medications that are substrates of CYP3A4 with a narrow therapeutic window, or strong inhibitors/inducers of CYP3A4 are ineligible, if they cannot be transferred to alternative medication. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * Patients with uncontrolled intercurrent illness * Patients with psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because BAY 1895344 as a DNA-damage response inhibitor, cisplatin, and gemcitabine may have the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with BAY 1895344 breastfeeding should be discontinued if the mother is treated with BAY 1895344 and for 4 months after end of treatment. These potential risks may also apply to other agents used in this study