The only weapons that can't hurt mines are pistols, snipers, redeyes, other mines, and MAD tank shockwaves. Unlike vehicles, mines are vulnerable to splash damage.
Mines self-heal 20 seconds after taking damage, until they become hidden again. They cannot be repaired by any tools; if they COULD be repaired by tools, it'd be possible to increase their health far higher than the intended amount - as demonstrated during Gamma where the fools in charge decided to make mines vulnerable to the Mechanic's wrench (yeah totally not allied biased). Any mine that had been wrenched was not only able to survive a V2 landing a foot away from it, but the mine wouldn't even be revealed!