Doom 3: runs regardless of install
Doom 1 and 2 from BFG disc: only run if the game is not installed to the HDD
Doom 1 and 2 XBLA versions: run regardless of BFG disc install, but you have to have bought them separately
I was on Minecraft for a while (new update for the XBL version today) so I'm just now starting a game up on D3. So far it's Doom 3. Same graphics, same intro areas and dialogue. The movement is definitely ripped straight from Rage. The weight of your character and the feel of looking around with the controller feels exactly like Rage.
The reason for the lack of co-op is because the original had no co-op. Vicarious Visions ported it to Xbox and changed a bunch of stuff around to make co-op work. for the BFG edition, id took the original PC code and ported it themselves, with some stuff from Rage (like the player movement). To include co-op they would have had to either rework a bunch of stuff like Vicarious Visions did, or use the VV versions of the levels. I can see why they didn't include co-op. I do wish they had, but I sort of get why they didn't. Using drastically altered levels wouldn't be faithful to the original. The biggest complaints I've heard about the BFG edition so far is that they didn't improve the graphics for Doom 3, but I never expected them to. I expected something true to the original, and that's what this is.
EDIT: They did say they updated the game to make it slightly more playable (which either means easier, or less annoying, depending on how you look at it). One thing they mentioned was more ammo lying around in better places. One thing I just noticed, and this may be a one time hint: I got to the first storage cabinet that you need a code to open, and a tip popped up telling me whose log to listen to for the code. EDIT: Nevermind, it only did it with the first one.