Precursors are Flood, according to Promethean.

To add to your point, Halo technically had a level progression in Halo 2. Battlefield 1942 had it in 2002. It's a staple of FPSes.

Guns aren't what necessarily make an FPS. It's how it is presented. On the point about guns, however, Halo 1 had the Assault Rifle start as main, with the "pro weapon" being a pistol. In Halo 2, the SMG replaced the AR, and the Battle Rifle became the pro weapon. Halo 3 brought back the AR, but it wasn't as powerful…