Background/Objectives: Combining heterocyclic fragments provides a strategy for developing compounds for exploratory anticancer evaluation. This study aimed to synthesize and characterize five thiazolyl-methylthio-1,3,4-thiadiazolyl-quinazolin-4(3H)-one compounds (5a–e), assess their properties in silico, and investigate the selected ones for cytotoxicity and their effects on irradiation-associated DNA lesions. Methods: Structures were characterized by IR, HRMS, and 1H- and 13C-NMR spectroscopy. Computational studies included physicochemical and ADMETox predictions, multi-platform target prediction, and molecular docking to A2A, Erβ, and CDK5. Compounds 5a and 5b were evaluated for cytotoxicity and cell-death profiles on one colorectal cancer (CRC) cell line (HT-29), two breast cancer (BC) cell lines (MCF-7 and MDA-MB-231), and one healthy fibroblast cell line (BJ), using cisplatin (Cis) as a reference. DNA lesions were assessed by the Comet assay in both breast cancer models before, immediately after, and 24 h after 4 Gy irradiation. Results: Compounds 5a–e were obtained in yields of 45.17–83.27%. Compounds 5a and 5b had lower 24-h IC50 values than cisplatin in MCF-7 cells, but higher values in the other tested cell lines. Compound 5b showed higher selectivity indexes than both 5a and cisplatin across the cancer models, although its values of 1.14–1.34 indicated modest selectivity. Cell-death profiles differed between treatments, with larger necrotic fractions following 5a or 5b treatment in MCF-7 cells. Both compounds were associated with higher post-irradiation lesion scores than untreated irradiated controls. Scores increased further at 24 h in MCF-7 cells but decreased in MDA-MB-231 (triple-negative) cells. Pre-irradiation scores were also elevated in treated MCF-7 cells. Computational profiling identified comparatively more favorable predicted solubility for 5a and 5b. Cross-platform ADMET comparison consistently indicated low Caco-2 permeability, CYP3A4 inhibition, potential liver-injury risk, and high plasma–protein binding, while several absorption, transporter, and safety endpoints remained discordant. Target predictions showed weak cross-platform convergence. Docking provided the strongest, although moderate, structural support for A2A binding by 5a–d. Conclusions: The findings support further investigation of 5a and 5b, particularly their effects on irradiation-associated DNA lesions. Priorities include improving potency and selectivity, validating in silico predictions, and evaluating radiosensitizing potential through clonogenic survival studies.