Synthesis, Spectroscopic Characterization, Density Functional Theory, Molecular Docking and ADMET Evaluation of Novel Carbazole-Based Pyrido[2,3-d]Pyrimidine Urea and Thiourea Derivatives

New carbazole-based pyrido[2,3-d]pyrimidine urea and thiourea derivatives (4–6) were synthesized from the previously reported amino intermediate 3, while the previously reported compounds 2 and 3 were included for comparative purposes. The new derivatives were characterized by FT-IR, 1H NMR, 13C NMR, and elemental analysis, and the available spectroscopic and analytical data were evaluated as supporting evidence for their proposed molecular structures. To gain insight into their electronic properties, Density Functional Theory (DFT) calculations were performed at the B3LYP-D3BJ/def2-SVP level of theory. The calculated HOMO–LUMO energy gaps ranged from 2.2711 to 3.3194 eV, revealing noticeable differences in the electronic properties of the investigated derivatives. The fluorophenyl-substituted thiourea derivative 4 exhibited the smallest HOMO–LUMO energy gap together with the highest molecular softness, indicating greater calculated electronic softness, whereas the dichlorophenyl-substituted urea derivative 5 displayed the highest electrophilicity index, indicating the greatest calculated electron-accepting tendency within the series. In contrast, the dichlorophenyl-substituted thiourea derivative 6 exhibited the largest HOMO–LUMO energy gap and the highest calculated hardness. The potential ligand–protein interactions of compounds 2–6 were further investigated by molecular docking against the epidermal growth factor receptor (EGFR, PDB ID: 1M17) and vascular endothelial growth factor receptor-2 (VEGFR-2, PDB ID: 3WZE). Compound 6 yielded the most negative docking scores within the investigated series toward both EGFR (−12.0 kcal/mol) and VEGFR-2 (−12.2 kcal/mol) under the initial docking protocol. The docking procedure was assessed by re-docking the corresponding co-crystallized ligands, with an RMSD of 1.64 Å obtained for EGFR. Additional flexible-receptor and repeated docking analyses against EGFR demonstrated that the minimum docking score was not necessarily the most frequently represented score among the generated poses, supporting consideration of docking-score distributions rather than reliance on a single top-ranked pose. In addition, in silico ADMET analyses using pkCSM and SwissADME revealed model-dependent pharmacokinetic predictions, including contrasting predictions of intestinal/gastrointestinal absorption for derivatives 4–6. Overall, the combined experimental and computational results provide structural characterization and comparative theoretical evaluation of this carbazole-based pyrido[2,3-d]pyrimidine series. The computational findings support further investigation of compounds 4–6; however, experimental biochemical and cellular studies are required to validate the predicted ligand–protein interactions and their biological relevance.