Regarding the HDD: If you're talking about a 7200 RPM drive, then yes, the newer HDD's are really quite fast compared to the older ones.
If you are talking about a 10'000 rpm drive (see below, I have one of those), then buy a cheaper HDD and spend the extra money on an SSD. They are too expensive for what they offer in my opinion.
But even though these HDD's are quite fast compared to older HDD's, they are noticeable inferior compared to an SSD (even the slower ones).
As an example: I'm running one of my rigs on a Western Digital 10'000 rpm Velociraptor, and even though that HDD is insanely fast for an HDD, the system feels rather sluggish compared to an SSD based one. It becomes especially noticeable when I'm running out of memory and have to resort to the SWAP file (memory page file in windows? not sure on that one).
Having said that, booting up and starting programs is noticeably slower, but not annoyingly so. You can still work on an HDD-based system without feeling like tearing your hair out

.
Regarding the CPU cooler: I've been using water exclusively for the past 10 years. A quick Google search has not led me to any horrible shortcomings of the candidates that have been mentioned, but I've been out of the air cooling game for so long I really should not be giving advice on that one. I know about a few closed loop water coolers and the Noctua air coolers, but that's about it :huh:
In conclusion:
If you never (or very rarely) run out of memory and have to resort to the swap partition or whatever its Windows equivalent is, then it is somewhat acceptable to install your System on an HDD if you're really not willing/capable to spend money on an SSD. It's still quite a good system.
But if you can in any way, shape or form afford even a small and comparatively slow SSD (Corsair, Samsung and Crucial come to mind, but as mentioned, my info might be slightly out of date), then go for that. It really is worth the money, imho.
If you have to prioritize: Keep the CPU cooler for now, and buy a 1 TB drive (not a very fast one, a WD 1 TB Caviar Green seems like a good choice), and a small SSD.
This should not be too expensive, and you can still upgrade the CPU cooler when you've saved up more money.
Of course, that's just how I would do it if it was my rig. Your priorities/preferences and needs naturally might differ from mine.
EDIT: The reason I mention Corsair, Samsung and Crucial are that they seem to be reliable last I've heard. Of course, Intel belongs to that group as well, but their drives usually carry a (sometimes hefty) premium. From what I've heard and read, just don't by OCZ, although I am not speaking from first-hand experience on that one.