The rapid integration of inverter-based resources (IBRs) is fundamentally transforming traditional power systems into power-converter-dominated power systems (PCDPS), creating unprecedented challenges for transient stability and protection. This paradigm shift from physics-defined synchronous generator dynamics to software-defined IBR behavior manifests as reduced short-circuit levels due to severely limited fault currents, software-controlled phase angles, suppressed negative-sequence injection, high rates of change of frequency (RoCoF), and complex multi-timescale dynamics. These developments challenge classical stability analysis methods and compromise conventional protection schemes. This comprehensive review examines emerging solutions across three domains: innovative stability methods to address multi-timescale dynamics, protection techniques resilient to IBR fault characteristics, and advanced controls to enhance system strength. We emphasize integrated stability-protection co-design, multi-domain co-simulation, and critical grid code evolution. Future research must prioritize scalable large-signal tools, grid-strength-robust GFM strategies, and cyber-physical resilience frameworks. Securing PCDPS requires holistic methodologies to transform inherent fragility into operational resilience.