Get more CD's/floppies and put Memtest86 on one of them (it's real easy to burn, use Nero or something and select burn a previous image), run it once and see if you get any errors.
Stick with the 250 SATA (find one with a 16Mb buffer if that one doesn't have one) usually Maxtor makes cheaper ones (price wise). Keep it at a gig of ram, 2 is just overkill. No need to upgrade the video card, looks fine as it is.
Look into eMachines (some, not all) they usually have very good deals and pretty good upgradeability. A mid level Dell would also be somewhat ok. With a 600$ price cap though you'll be hard pressed to find a machine that can play the newer games.
Use the newest drivers and see if they indeed go to 1600 x 1050. If not go back down to a set of older drivers. If they also don't do 1600 x 1050 it's because the monitor can't do it, not the drivers.
Opening the case of any computer bought from any company pre-built with a warrenty automatically voids it unless an authorized service technician opens it and validates that they themselves performed the work.
Working on a machine now (compaq Deskpro 6600) that has an unusual problem. The computer boots in safe mode fine but when booting in normal mode the screen freezes on the windows98 boot screen but continues to boot as normal (you can hear it in the background as it makes the normal windows...