Honestly HP machines are such junk that by the time you get the paste off, reflow it, see if it works, and sell it, you'll have spent more time, money, and effort than its worth.
To make matters worse, the HP a6600f is based on the NForce 600 series which is notorious for having a super hot north bridge that would burn out. It may not be a soldering issue, it could just be completely dead. The entire northbridge put out as much heat as some CPUs.
If you're doing this for a fun project then it's a great place to start because all you've got to lose is a 6-7 year old piece of crap that may not work anyway. If you're doing it for the money, then don't even bother because it's not worth it in the slightest.
To get the paste off you'll probably need a strong solvent. I recommend electronic contact cleaner. You can get some different types but the ones that are HCFC based are the best; unfortunately they're being phased out soon so you may not be able to get them.
As for the reflow project, you're going to need to keep the board completely flat which means putting it in a pair of vice grips if possible. If you don't, the heat gun will warp the board and only make matters worse.