I think the problem is that you're not paying careful enough attention to the numbers. You've referred to your native res at "1600x1050" and "1650x1080," when in actuality it is "1680x1050." It wouldn't surprise me if you're being equally careless with ingame resolutions. I am willing to bet that the resolution you are seeing in Doom 3 is actually "1600x1200," which is not a widescreen resolution, and it's also not going to work on your monitor (it is 1200 pixels tall, and your monitor is 1050 pixels tall, so it just isn't going to fit).
Here is what I suggest you do.
Commit this fact to memory: Your monitor's native resolution is "1680x1050." This is the resolution of your monitor, so this is the resolution that you want to get your games to run at. Make sure you remember the numbers exactly - you should be able to type "1680x1050" in your sleep. Getting a single digit wrong is likely to make the difference between a game working perfectly and a game not working at all.
Make sure you have the latest drivers for your video card, downloaded from ATI.com (never use drivers from your board manufacturer; throw away the driver disc that came with your video card)
Use a DVI cable to connect your video card to your monitor, if you aren't already doing this.
Any time you play a game and your native resolution (that's 1680x1050) doesn't seem to be available, look that game up on the
MGL. Some games require you to edit config files or do other crazy things to get widescreen resolutions working, and the MGL will tell you which games have special solutions, and how to use them.