check your processes (ctrl+alt+del) see if anything is using way to much cpu or ram,
Secondly for your virus scan, Installing when you have a virus is pretty much useless,
Try the symantec online virus scanner
[Only registered and activated users can see links. ]
For the shutting down, Do you know if any hardware is bad? I'm thinking it could be the psu or mobo or ram...But thats pretty weird...
What graphics card do you have? (It could have to do with the reboots)