Assuming you're gaming, that build will destroy anything you throw at it for the next 5-6 years. The 5970 is powerful enough to play any game with max details using Eyefinity (3 monitors as one, or 5760x1080 resolutions). The X4 955 is a little weak compared to the i7-930, but it's also nearly half the price, and it at least has an upgrade path.
To be fair, the other ones still a beast. The 5870 is powerful enough to play everything at max details with a single 1080p monitor. However, it will likely need to be upgraded/Crossfired in 3-4 years. The i5 is also a more powerful CPU, but there is no upgrade path for it, so if you ever need more CPU power, you're looking at a new build.
The cooling will be fine. The HAF 922 IS a full tower case, it's just mislabled a mid tower. The case is actually larger than most full towers (the Antec 1200 included) and is one of the few cases that will actually fit an HD 5970 in it without squeezing or modding.