I have a Core 2 Duo, overclocked to 4.5 GHz.
Kingston DIMM DDR 2 677 MHz RAM - 2 GB.
GeForce 7600 GS 512 DDR3 video card... everything runs great, no matter what I install. The only problem is that I have no internet at home, and at work I have:
Pentium D915 - 3000 MHz
The same RAM as at home, but only 1 GB, and a Radeon X1600Pro 256 MB video card.

With those specs, what problems could you possibly have? But the lack of internet is a real issue. Tell your parents that buying an expensive car without an engine won't make you happy. Bribe them with an ADSL modem and get unlimited internet; it's a great thing.
I wrote this without seeing your edits, but I'm telling you, unlimited ADSL is awesome; you can do everything.

Added 9 minutes later

By the way, don't think I'm bragging; I just saved up for this computer for six months.

We already thought so, my friend. In another month, you'll have internet not only at work but also at home. The market is such that everything new and the latest costs much more than it's worth, so it's better to buy hardware that's not the latest but is still quite good, from 2-3 months ago. You'll save a lot, and in a few months, both your current hardware and the "latest" hardware you wanted to buy will be equally outdated. That's it.