Wolfenstein came first, Doom followed from the same team. While some doom clones at the time continued to use the Wolfenstein engine it was fairly brief. Rise of the triad is I think the most impressive one using basically the Wolfenstein model?

Catacomb 3D (also by id Software) came first. Wolfenstein 3D was a further development of the same tech that happened to be much more popular, hence why people mostly remember that.

As I recall there were a few games using a similar engine at the time, it's just that they didn't left much of an impression. Some examples would be Blake Stone and Corridor 7.

Depends where you want to draw the line. You could trace the lineage all the way back to Rogue with its grid map where the player is fixed to the grid and moves in full block increments. Then you had those 3D maze games where you were still on the grid and stuck moving in cardinal directions, but you got a 3D representation of the area. Then Wolfenstein allowed the player to move freely on the same grid map. Finally Doom (finally!) dispensed with the grid and made the map out of vectors.