Safe quadruped locomotion over hazardous terrain cannot be reduced to selecting the locally safest foothold, because a geometrically reachable contact may still deteriorate the subsequent support phase, increase actuator demand, approach friction limits, or leave insufficient clearance for the robot body. This work presents STEP-CBF+, a reduced-order predictive foothold-and-posture screening framework that couples terrain-risk memory, DCM/ZMP-based support prediction, discrete-time barrier screening, energy-aware motion regularization, three-joint torque feasibility, friction-cone constraints, and anticipatory body-height adaptation within a single candidate-selection layer. The study considers a 13.5 kg quadruped with 3-DOF legs executing a crawl gait with duty factor 0.75 and evaluating 49 foothold candidates at each decision. Six hazardous-terrain conditions are examined over 25 randomized terrain and payload realizations and compared with nominal, risk-only, and reduced-order convex-MPC strategies. Performance is quantified using independent fall and slip outcomes, distance before failure, dynamic support margins, tracking accuracy, actuator and friction utilization, computation time, and component-wise ablations. Across all scenarios, STEP-CBF+ reduces the mean fall rate to 4.0% while increasing the stable-step fraction to 77.3%, compared with 16.7–18.7% falls and 72.2–73.7% stable steps for the three comparators. A particularly revealing result appears on inclined terrain, where stable time reaches 86.0% and the minimum dynamic margin improves to -0.032 m while peak actuator utilization is reduced. In the confined duck-under scenario, anticipatory posture adaptation enables collision-free traversal and zero falls, whereas all non-height-adaptive strategies fail. The method retains a mean selection time of approximately 13.6 ms, compared with 30.5 ms for the reduced-order convex-MPC implementation. The results indicate that the main benefit of STEP-CBF+ emerges when contact feasibility must be evaluated together with the consequences of the next support phase and body posture, providing an interpretable intermediate safety layer between terrain perception and lower-level locomotion control. The conclusions remain limited to reduced-order randomized simulation and do not constitute a whole-body nonlinear invariance proof or hardware validation.