dbrupp,
From the log, nothing stands out as far as mal-ware is concerned. Please keep in mind that doesn't mean your computer is guaranteed to be 100% clean from infection.
The only thing that even looks to remotely be a problem is the computer looks to have two AV (anti-virus) programs on it. I noticed, both, AVG and Microsoft Security Essentials installed. Having multiple AV softwares on the same machine can often cause some conflicts of random types. While I don't know if this could be causing the desktop icon problem, it's still something you may want to look into changing. I advise you disable/uninstall one or the other through Start > Control Panel > Programs and Features.
It's up to you which one you disable/uninstall (again, the main thing is we would like for just 1 of them to continue running), personally I would probably ditch the AVG unless you paid for it. Microsoft Security Essentials is one of the best AV programs and it's entirely free!
I will keep researching the icon issue, but at this point I have no other suggestions to offer.