Three-dimensional (3D) bioprinting enables the fabrication of scaffolds with highly controlled architectures; however, the extent to which modest shifts in bioink composition simultaneously influence scaffold stabilization and the resulting early cellular microenvironment remains insufficiently understood. This study investigated the influence of hydroxyapatite (HAp) concentration within a low loading range (1–2% w/v) on the printability, crosslinking stability, and microarchitecture of pre-crosslinked alginate–gelatin composite bioinks, as well as on the immediate viability and morphological adaptation of encapsulated human dental pulp stem cells (hDPSCs). Bioinks were characterized via physicochemical analysis, while printing performance was evaluated under varied operational conditions. Structural properties were assessed through micro-computed tomography (micro-CT) and swelling kinetics, whereas early cellular responses were monitored via Live/Dead staining and fluorescence-based morphometric analysis. Printability parameters were primarily dictated by nozzle diameter and layer count, with the 22G nozzle yielding larger pore sizes (1,132 vs. 940 µm) and pore areas (0.749 vs. 0.515 mm²) than the 20G nozzle. Micro-CT confirmed highly interconnected structures with porosities of 74–78%, where the 2% HAp group promoted a highly homogeneous trabecular spacing centered between 700 and 850 µm. Incremental HAp content significantly suppressed matrix swelling kinetics in a concentration-dependent manner. Biologically, all formulations sustained high post-printing cell survival (>70%) that exceeded 90% after seven days. HAp-containing hydrogels promoted significantly higher cell viability and increased cell spreading areas. These findings underscore that small variations in HAp content influence scaffold stabilization kinetics and early stem cell behavior, while manufacturing parameters remain the primary determinants of initial geometry.