How do you know when a cpu is damaged?

At best, it just wont work,
at worst, it could short out your Mobo and possibly other components would be affected.
Thermal paste, especially conductive pastes, should not be anywhere other than on the top of the chip/bottom of the H.sink and a microscopic amount there at that.
You can clean it off using high alcohol solutions.
Moto

When you get a HARDWARE that is too powerfull for your CPU ... it cause dammage s ...LIKE geting new VGA card .. HEAT can kill your CPU ...keep your CPU cool enough ..heat slows hadrwares donw .. OVERCLOCKING your CPU CAUSE too much damage to it .. LOW PSU for CPU ...doesnt let CPU to work well & some times it Cause damages to it .. there are some reasons which can cause damage to you CPU ... KIDS can Cause Damage to your CPU & PC too lol

A damaged CPU might throw errors, it might make the computer behave flaky in general, it might do 2+2=5 which would screw up many programs, or it might not boot up at all which is more likely. In general its pretty hard to damage a CPU quickly without doing some heavy OCing with high voltages.