If one's amp would be clipping, resulting in a blown sub, is it a permanent issue? Or does the gain, bass eq, and other filters need to be tuned to insure that it won't happen with the new sub?

Clipping is transient, as is power output. And it can be eliminated.

The amplifier is being overdriven on the peaks of the music, and as a result the top and bottom of the waveforms are cut off (which is why it's called clipping, the wave looks like the tops and bottoms were clipped off). This is bad because while this is occurring, the average power of the signal increases (it can double in the worst cases) which can overdrive the connected speakers.

Yes, it is primarily a function of an improperly set gain, bass boost, boosting the signal with an EQ, overdriving the input of the amplifier with too much signal, etc. If you were experiencing clipping, it was likely a result of one of these being improperly set and reducing their level will alleviate the problem.

How exactly were your system settings adjusted? What amplifier and headunit are you using?