Ok, I'll see if I can put this to rest:
1. Chris Vrenna's band "Tweaker" sounds absolutely nothing like Tool. I'd describe his stuff as atmospheric, industrial pop/rock... not one track has ever sounded like a Tool song. You can download a few tracks at Tweaker.net and see for yourself.
2. I'm a big Tool fan. The Doom 3 theme is DEFNINITELY Tool, remixed/reworked by Vrenna (who has remixed tracks by A Perfect Circle in the past). It's a reworked version of the songs Lateralus and Third Eye. There are actually very short and somewhat distorted VOCAL samples (definitely MJK from Tool) which begin at 1:08 in the Doom3 theme. In fact, between 1:28 and 1:34 you can hear MJK's signature whiney growl (somewhat chopped up). A dead giveaway for any Tool fan. That's my opinion... believe it or don't. But I will eat my own head if that isn't Tool.
3. A Perfect Circle and Tool are not the same band, by the way. Musically (not vocally) they sound nothing alike... it's the vocals that are similar due to Maynard's presence in the band. No other members of Tool have ever played with APC, and vice versa.