Whoa there, you need to learn your history young one. While Heretic and Hexen were indeed imporvements over the Doom Engine, Quake was the first major Fully 3D FPS. It was a completely different engine. Also Duke Nukem 3d, as awesome as it was only a little more advanced than the Doom engine. In fact the most advanced 3d engine until Quake was the engine used by System Shock and Ultima Underworld. The environments in those games were actually completely 3d, something that the Doom Engine could never pull off, and the Build engine could not achieve until its Shadow Warrior version.
Agreed, though I think you are ignoring what QuakeWorld and Quake 2 did for PC online gaming. UT added a LOT, and was badass as hell, but it was nowhere near as influential as those early quake builds.
Debateable, Half Life 1 and 2 I think have vastly overrated stories, in fact I don't think HL2 REALLY gets compelling until EP2 (story wise.) HL1's story was ok at best. System Shock 1 was doing a deep story in a shooter/RPG long before Half Life did. Half Life was just a LOT more popular.
I also think that you are ignoring some of the major advances that the Halo and Call of Duty franchises have brought to shooters. Halo had unparalleled AI and Outdoor areas, and vehicles that weren't matched at all that generation, and Call of Duty did war better than anyone else. Battlefield 1942 is also deserving of mention for bringing huge multiplayer sessions into vogue, and Planetside deserves mention for being the first MMO shooter.