Reguarding the video card questions..
That is the king of AGP. That is the best card you can buy. Err, I mean chipset.
That is better then most Pci Ex cards. Can it last a few years, yes. But it wont be by any means topping any new Pci Ex video cards. But for todays games, it will get the job done, and later games it can handle, not at max settings though.
I personally say its worth it to get Pci Ex. But with that you need new ram and processor and probably hard drive. Ugg, that would be a pain in your ass. So to keep this short, sweet and two the point, get that 7800GS and have the best AGP card that money can buy.
Can I replace my motherboard, keeping the same processor and other parts in my computer without losing any of my saved hardware and information
That depends on socket. odds are good that you wont be able to get Pci Ex, and keep the same processor, and hard drive. Although, you can easily back up your hard drive. Or have someone do it for you. Hell, id do it for free for you.
I am looking for suggestions for a 1) Motherboard, 2) Video Card, 3) Monitor that will all be compatible. I dont want to spend more than 350 for a video card or more than 350 for a monitor, price of the motherboard doesn't really matter.
Lets see, I know I didnt answer all your quetsions- Sorry, its getting late here. Motherboard, that all depends on what you want to do. 7800GS for a video card if you are going to keep your current set up, and a 7600GT if you are going to get Pci Ex. Monter just response time under 12ms.
So, we can work out the fine details later, but overall, it comes down to how much cash you want to spend, or effort you want to put in this computer.
Like I said, keep it short and sweet, and get a 7800GS, or do it lengthy and pricy, and your looking at new processor, ram, videocard, and maybe hard drive.
Tell us what you feel like doing, and when we will hook you up with everything you need to know.