I know that a 2.4 Ghz, 512 MB Ram, NVidia GeForce had no trouble running GC2. Considering how many beta testers have ATI cards, you should be alright to run GC2.
Your problems with Civ4 are definately due to (for it) slow processor, low RAM (it needs a full Gig), and the fact you have an ATI card. Upgrading your ATI's drivers to the latest should/might fix your black terrain issues. As for your other issues... well, faster CPU and more RAM memory is what's required. It's irritating, but the game's listed recommended specs are too low to meet it's actual minimum. Too bad for Civ fans.
P.S. Yes, you get in soon. I don't know if you'd have immediate access to the current beta (4b), or if you'd have to wait for the next build (4c). But you'd certainly be granted access quickly, judging from past Star Dock performance on GC2.