Since the rubbish emails were sent to your contacts, chances are your email account has been hacked, so change your password and choose a strong one, and while you are at it check that the email address or phone number to be used if you forget your password is correct and hasn't been changed.
If you have a look in your sent folder, you might find the rubbish emails in there, which will confirm that the account was hacked.
Now, I don't use Facebook, so can't say for certain, but I know a lot of social networking sites ask you for your email address and password so they can show you if any of your contacts are already on that site. Facebook may well do the same. The idea is to list your contacts as potential "friends" or to email them in your name inviting them to join the site. I never provide that information, and tend to get a bit annoyed by the emails supposedly from friends who have been silly enough to do so.
If the emails sent genuinely were rubbish and not invites to Facebook, it's unlikely that Facebook sent them - it's not in their interests to spam the contacts of their members. So I'd guess that the email address you used to sign up again was one which is already out on the net somewhere and was targeted to be hacked.
If you want to double check the results of the AVG scan, Kapersky have a security scan program which you can download from
http://www.kaspersky.com/security-scan - it will only detect and report on any malware and not remove it, and it won't conflict with AVG. If it finds anything, you can get their virus removal tool from
http://www.kaspersky.com/virus-scanner and you can delete either or both from your computer when you have finished with them.