Hi,
Windows 2000 (and NT and XP) are totally different operating systems compared to Windows 95 and 98. One big change is that Windows 2000 doesn't allow programs to directly access hardware.
In the bad old days before more powerful hardware and DirectX, many programmers wrote their own routines to do direct manipulations on the graphics cards to squeeze every last drop of performance out of the machines.
If this is the case for your games, then its possible they wont ever be able to run under Windows 2000 no matter what you do.
You will be able to upgrade to Windows XP which has built into it a "compatibility mode" with the older operating systems which supposedly overcomes some of these problems with comprehensive "Virtual Device Drivers".
I've found that some programs wont install on WinXP (in my case) but will happily run there without having to resort to compatibility mode - it was simply a check in the install process that disallowed installation under anything but Win 98/95
So what I've done for myself is to create a dual boot. One option boots to Win98 and the other to WinXP.
Here are some good instructions for doing that if you want to tackle your problem that way.
http://www.microsoft.com/windows2000/techinfo/administration/management/mltiboot.asp
(remove spaces from the above URL)
Cheers,
Tim.