I run Toad almost daily over VPN, sometimes for 12 hours at a stretch. Other than the usual VPN issues that drop my Oracle connections after inactivity, I’ve seen nothing like what you’re seeing. I’m on Win7-64 running Toad 12.6.0.53.
Oddly enough, apps that require Adobe Flash crash on me over VPN, so I don’t want to totally discount Toad being related to your issue, but from your description, I’d run a thorough RAM check first. Lookup “memtest86” (I use this on every new PC) and I’ve also read that “mdsched” will also do the trick. Both will reboot your PC to run the program outside of normal Windows.