[Resolved] Malware (url.adtrgt) issue / HJT log included

It's odd that the stats say that my dad's PC is running faster because it's definitely slower than my sister's laptop.

Are you referring to download speeds, or general running speeds? Your father's PC shows a better internet connection speed, and that is possibly due to it having a cable connection.

The one thing that you don't have installed and that you should have is a firewall. You have a version of one built into your router in the form of Network Address Translation, but it doesn't offer the fullest protection. I like to have a PC clean before installing one so I didn't mention it before, but I don't think that your issues are malware related.

There are a few free firewalls available.
Comodo Firewall Pro, available here. This download has both a firewall and anti-virus in the same package, so be sure that you uncheck the AV option if you choose to install this one.
PC Tools Firewall Plus, available here.
Online Armor Free, available here.

It is important to note that you should only have one firewall installed at a time, but you can download them all to your Desktop and install each in turn to see which one you prefer.

Understanding and Using Firewalls: http://www.bleepingcomputer.com/tutorials/tutorial60.html

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Given that there is nothing that has identified anything nasty on your system, I can only assume that any system slowdown is due to either the age of your computer(s) or the Operating System installations. Unfortunately installing and uninstalling programs and Windows Updates can both contribute to a slower system over time, and the best thing that there is for this is a fresh install.
XP is at Service Pack 3, so your system is behind the times there and Internet Explorer is at version 7, so that really needs updating too.
I suggest you seriously think about backing up and then reformatting and reinstalling your Operating System and then fully updating it as this will give you a fresh start and also a more secure system.

I was referring to running speeds because my sister's computer runs a lot faster, especially on the internet. I will go and download one of the firewalls and see if it helps to speed up the computer. I mentioned earlier that Firefox seems to run faster than IE. I recently downloaded the newest version of Firefox and I didn't like some of the new features, so I changed the CSS code and installed some add-ons. Could this be part of the problem?

Use another browser and compare surfing speeds.

I mentioned earlier that the internet did seem to run faster (a little speedier, but not lightning quick either) on IE than FF. Do you think changing the script has a part in the slower speed in FF? Then again, the internet's been slower since the whole malware issue, so I guess it's difficult to determine what the cause is.

Having looked and relooked at all the information that you have posted, I can say that neither I, nor anyone else who took the time to check, can see anything malicious in anything we looked at. That doesn't guarantee a malware-free computer, but it is a good indicator of cleanliness.
I could speculate as to the cause of the slowness, but that is exactly what it would be - speculation. My best advice, because it is the quickest and easiest, is to reformat and reinstall because that covers most, if not all, of the potential causes. Hunting for the proverbial needle in the haystack could lead you to a solution, but it could also not.
As the problem isn't malware-related, there is very little else I can do for you, so if you wish to persevere with finding the cause of your problem, I suggest you post here: http://forums.whatthetech.com/Microsoft_Windows_f119.html - they will hopefully have more of an idea than me.

Alright then, thank you very much for your help. I appreciate you sticking it through with the situation. At least the silver lining is that the computer seems to be malware-free! Thanks again! You've been a great help.

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