I agree with EntityAnomly. It seems like this is an overheating issue.
The PSU should be fine, the 3800x2 takes 110W in its worst case scenario, and the 9600XT and HDD's shouldn't be overtaxing the PSU with 250W.
Where is your computer located? If its in some enclosed space, move it out. If that doesn't work, take off the side panel and see how that goes. You have, hopefully, updated anti-virus software and a firewall so worms and viruses shouldn't be affecting things.
When your system is slow, check your task manager, if some process is sucking up your processors, it should show in there.
Double check your BIOS settings - I've seen some ASUS boards don't recognize the processors correctly and because that happens, its severly underclocked.
Your HDD should be ok. Its true that HDD technology has made leaps and bounds, but seriously, your 30GB should have ATA100, I hope. If its ATA66, you definitely need to get a new one or use your other HDD. And if its 5400rpm, you should only use this drive for document storage.
You also said its slow on reboot. Is that a cold or hot reboot? Cold = computer is absolutely in its off state. Hot = you pressed the reset button.
If you're running Cool 'n Quiet, have that up to see where your processor's speed is.
Make sure you seated the CPU fan correctly and you used the right thermal paste - those make all the difference in the world, plus it might save your CPU from meeting a firey doom.
Also make sure Norton plays well w/ the ATI drivers. you may have to do some research online for that. Norton IMHO is a resource hog and has alot of conflicts with different softwares.
If things are happening absolutely randomly, check your PSU. It may be bad (altho its an Antec). Might want to snag a PSU checker from your friends or buy one.
If the PC slows after being powered on for a while, have you checked to make sure you're not overheating?
Hotter = Slower
I know you've stated that it is slow to boot up as well, but is that all the time or is that when the PC has been on a long time and is potentially "too hot"?
I have a friend who had a lot of problems with his 9800 AIW causing some of the same issues you are having so I would say that Schmide's idea would be another good way to go.