Sounds like you have multiple problems, and at least one of them seems to be the HDD is giving out. You probably also have a pretty serious malware problem which doesn't help matters any.
Of course if I'm right, and the HDD is giving out, then that kind of solves the whole formatting issue for you. And you'll want to get on making sure you have restore media for this system post-haste. Not to mention back up any important data.
At four years though, with an HP, I'd say you've already exceeded every expectation you could have possibly had for the thing. Yours is a system from the height of the Fiorina era at HP, which saw the company locked in an epic struggle for supremacy with Acer, except that they were, and still are, fighting over which company can build the poorest quality system. Acer keeps vacuuming up brands that fail like Gateway and Packard Bell, HP has already effectively killed off the Compaq brand, so now they can focus on driving their own brand into the ground.
But I digress. Given the specs on the system, that it's 4 years old, and you have (at a minimum) what appears to be a failing HDD, I'd say it's time to consider a new system. Odds are the HDD is just the most visible sign of much deeper issues.
If you have some kind of sentimental attachment to it, then at a minimum you're likely going to need a new HDD. Then you'll have to contact HP and hope they'll be able to supply you with restore discs. If not, then you may want to consider adding some additional RAM and a copy of Windows 7. If the integrated graphics could handle Vista, they should be able to handle 7, so there's at least that. But when it's all said and done, if you can't get the restore media off HP, you'll probably find you're going to be crossing the 50% mark of the price of a new system with better specs, at which point it quickly becomes less and less economical to repair your current system.
If you do buy a new system, my advice would be to avoid HP, Compaq, Acer, Gateway, and eMachines systems like a leper with the plague. I'd also avoid pretty much any laptop under $1,000 since you tend to hold onto them for long periods of time. You get much below $1,000, and there's not a lot of profit margin to be had in the system. So just stop and think for a minute. If you're running a business, and you're making maybe $10-$20 profit on some product that retails for say $500, how much time and energy are you going to put into making sure this is a quality product? Probably very little. You're going to save your efforts for those things that actually bring in a decent amount of money. That's laptops starting at about $1,000. I'm not suggesting you necessarily buy an Apple laptop, but if you look at their prices, they tend to start at about $1,000 for the basic entry level MacBook. Apple is a company that doesn't tend to waste its time on loss leader products. If they can't make a respectable profit on something, they don't do it. If Apple can't do it, odds are HP or Acer can't do it either.
Gotten off on a bit of a rant, but basic point is that if you buy cheap, you get cheap. Good general rule of thumb to keep in mind. Applies almost universally to all electronics.
So, formatting will be unlikely to really help if you have a failing HDD, which the intermittent nature of some of the slowdowns strongly suggests is going on. Meaning you need to get restore discs from HP, if possible. Given the age of the system, and it's rather low end specs even for 4 years ago, you'd probably be much better off just buying new. If you buy new, be sure not to be stingy, or you will suffer the consequences. If you want to fix it, I'd suggest adding at least another 1GB of RAM along with the new HDD, and if you can't get the restore media off HP, consider buying a copy of Windows 7. Or Linux is always an option to keep costs down.
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