About 2 months ago I played a few painkiller demos and remember one of them having a lot of locked doors that only opened if I killed all the enemies in the area. Is this full game like that? The movement is really fast so I'd like to run through levels as much as I can like in DOOM.

I played a little of it earlier and yeah, you have to clear some areas before you're allowed to progress any further.

Nearly every level is like that, actually. The core gameplay is killin' folks.
There is a way to simply blow up all enemies if you want, however. Cheat is "pmexplodeenemies."