Testing the Safety of the Combination of Anti-Cancer Drugs CX-5461 (Pidnarulex) and Trastuzumab Deruxtecan (T-DXd) for Human Epidermal Growth Factor Receptor 2 (HER2)-Positive Solid Tumors and Breast Cancer

Inclusion Criteria: * DOSE ESCALATION PHASE ONLY: Patients must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective * DOSE EXPANSION PHASE ONLY: Participants must have histologically or cytologically confirmed invasive breast cancer, with either locally advanced or metastatic disease * Age ≥ 18 years. Because no dosing or adverse event data are currently available on the use of CX-5461 (pidnarulex) in combination with T-DXd in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 (Karnofsky ≥ 60%) * Absolute neutrophil count (ANC) ≥ 1,500/mcL * No administration of granulocyte colony-stimulating factor (G-CSF) is allowed within 1 week prior to screening assessment * Platelets ≥ 100,000/mcL * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * Total bilirubin ≤ 1.5 × institutional upper limit of normal (ULN) (or ≤ 2 × institutional ULN in patients with documented Gilbert's syndrome) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) ≤ 3 × institutional ULN (or ≤ 5 × ULN in patients with liver metastases) * International normalized ratio (INR)/prothrombin time (PT) and activated partial thromboplastin time (aPTT) ≤ 1.5 × ULN * Glomerular filtration rate (GFR) ≥ 60 mL/min/1.73 m\^2 (using the Cockcroft-Gault equation for participants with creatinine levels above institutional ULN) * Patients must have had at least one prior line of cytotoxic chemotherapy. Patients can have received an unlimited number of additional lines of chemotherapy, targeted therapy, biologic therapy, or hormonal therapy. Patients must not have progressed on a prior anthracycline in the metastatic setting. Receipt of anthracycline in the (neo)adjuvant setting is allowed, provided that disease recurrence occurred later than 6 months after the completion of treatment * Prior poly (ADP-ribose) polymerase (PARP) inhibition is allowed * No specific germline mutation is required * DOSE ESCALATION PHASE ONLY: * HER2-positive (IHC 3+) solid cancers, * HR+ HER2 positive/low/ultralow breast cancer or triple negative breast cancer (TNBC) HER2-low breast cancer, with HER2 positive defined by the current American Society of Clinical Oncology (ASCO)/College of American Pathologists (CAP) guidelines, * HER2-low, with HER2-low defined as IHC 2+/in situ hybridization (ISH)-, IHC 1+/ISH-, or IHC 1+/ISH untested * DOSE EXPANSION PHASE ONLY: * Either the primary invasive tumor and/or the metastasis must be: * HER2-low, with HER2-low defined as IHC 2+/ISH-, IHC 1+/ISH-, or IHC 1+/ISH untested. Any estrogen receptor (ER) and progesterone receptor (PR) expression is permitted but must be known, or * HR+ HER2-ultralow defined as IHC 0 with membrane staining * DOSE ESCALATION PHASE ONLY: * Participants must have evaluable or measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 * DOSE EXPANSION PHASE ONLY: * Participants must have at least one lesion that is not within a previously radiated field that is measurable per RECIST v1.1. Bone lesions are not considered measurable by definition. Biopsy of the lesion that will be used for disease evaluation (measurable disease) is not allowed in the dose expansion portion of this study. * Peripheral neuropathy grade ≤ 1 * Patients must have left ventricular ejection fraction (LVEF) ≥ 50% by either an echocardiogram (ECHO) or multi-gated acquisition (MUGA) scan within 28 days before randomization/enrollment * 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 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 II or better * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * The effects of CX-5461 (pidnarulex) and T-DXd on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control or abstinence) 14 days prior to study entry and for the duration of study participation and for at least 7 months (women of childbearing potential \[WOCBP\] only) after the last dose of study drug. 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. Women should not breastfeed while taking CX-5461 (pidnarulex) and T-DXd and for 7 months af