Is it possible after a virus attack, if I am not sure whether all the bad files have been quarantined and deleted, to get back to a clean, non-infected earlier state by using System Restore?

I ran AVGFree, and also TrendMicro's Housecall, and think I may have eliminated all trace of the virus. But I am not fully comfortable, as the computer now runs slow.

Any suggestions?

If your computer was infected, then System Restore was probably also infected.

You say you're using AVGFree and TrendMicro's Housecall.
You should not have 2 Antivirus programs running at the same time.
This will cause conflicts, and possibly cause the slow down you're talking about.

Trend Micro's Housecall is an online scan. I'm guessing that is the one Chatice is referring to; if it was run after scanning with AVG it would not directly be the cause of a system slowdown. tg1911 is correct in stating that System Resore could still be infected & you would need to remove them. If your computer is still runnung slow I would recomend posting a Hijack This log. At the top of this page, click the Start Here! link next to "Do you have popups or other malware infecting your computer?".

Hey do not mess with it let's see if you are infected and we have a team of experts that will troubleshoot your problem and help you thru all of this for the gigantic price of = FREE!!!!!!!!!!!

I suggest you post a HijackThis log for examination.
A member of the HijackThis Team will walk you through, step by step, how to disinfect your computer.
Once you post your log, don't make any changes to your system, as that could change the results of the posted log, making it difficult to properly clean your system.

Read Preparation Guide for use before posting a HijackThis Log.
http://www.bleepingcomputer.com/forums/topic34773.html

Please read, and follow, all directions carefully!!!

Then, run a log, and post it in the HijackThis forum, at this link below.
http://www.bleepingcomputer.com/forums/forum22.html Do not, fix anything, yet.
A member, of the HJT Team, will help you out.
It may take a while to get a response, because the HJT Team are very busy. Please, be patient, as these people are volunteers. They will help you out, as soon as possible.

NOTE:
Once you have made the post, please, DO NOT make another post in the HJT forum, until it has been responded to by a member of the HJT Team. The first thing they look for, when looking for logs to reply to, is 0 replies. If you make another post, there will be 1 reply. The team member, glancing over the replies, might assume someone is already helping you out, and will not respond. So, just make your post, and let it sit there, until a team member responds. This way you will be taken care of, in the most timely manner.

Thanks for the suggestions. I have posted my HijackThis log on the forum, and await the HJT team's guidance.

Besides the slowing down, the other problem (or annoyance) is the popup in the system tray that I can't remove. Every few minutes it shows up in a red framed box telling me that "Your computer is infected" and offering to sell me SpywareQuake.

You definitely have a SpywareQuake infection.
That will slow down your system.
A member of the HJT team will be able to help you remove it.

And as ThorXP stated in his post: