can you just update the summary
1) The laptop restarts frequently
2) The autorestart box is unticked
3) you get NO Blue screens
4) The PC works fine in safe mode and does not restart
5) UBUNTU will run from the CD when booted off your sister in laws PC - BUT NOT your PC
6) You have cleaned all the fans and vents
can you add a few points if missed and confirm the above
If it works perfect in safe mode without restarting - but not in full mode.... Then we need to look at the drivers and programs that are used on full mode..
you could try a clean boot and see what happens
http://support.microsoft.com/kb/331796