How plausible is it for the lunar body of the home planet of the whole Klingon empire be "their key energy production facility" with lots of mining operations?
Very, I'd say - they are a paranoid empire, so they would want to keep a key supply close to their home even if there were much richer pickings in distant star systems. And it's unlikely that they could mine an entire moon empty of its key resources within just a couple of centuries.
It might also be that no culture can become a star empire unless it has certain key materials available at its home system. Perhaps Praxis blew up because it contained energetic minerals necessary for warp drive (dilithium might fit the bill, although opinions vary)?
Although perhaps it was because Klingons had stacked it full of powerplants making use of imported fuels (and used those for refining the local materials). We can also take "overmining" literally and say that their security department deployed far too many land mines to repel intruders, and caused a chain reaction when a Romulan or Federation spy stepped on one of those.
And would its destruction really threaten to deplete Kronos's ozone layer in fifty years?
We don't know if it's a case of ozone depletion: Spock's words were "deadly pollution of their ozone". Ozone is a deadly poison, so possibly the planet would be polluted by excess ozone because of the explosion. Might be "their" ozone from the upper atmosphere (where it actually does some good), forced down by the calamity.
Certainly an energetic phenomenon would
create ozone out of oxygen where it's needed the least, in the lower atmosphere (just like ordinary cosmic radiation creates it in the upper atmosphere but fails to penetrate deeper down). Whether that would still be "their" ozone is a case of semantic wrangling, to make sense of jargon the writers themselves clearly didn't quite understand...
For comparison, how plausible is mining on our own moon, and what if it exploded?
Luna is great for obtaining light metals and oxygen, which are both quite useful for space industries. It can also be used for harvesting exotics such as cosmic helium-three, which is trapped in lunar soil and easily extracted but is a pain to gather here on Earth, what with the atmosphere being in the way. But Star Trek level industries might be more interested in heavy metals, and would be able to mine gases from gas giants.
The Moon exploding would be a problem in terms of the explosion itself; the mere fragmenting of our natural satellite would matter little, with a few boulders semi-harmlessly impacting Earth but most no doubt just forming an orbiting cloud that gradually ground itself into nice dust rings. Today's artificial satellites would hate that, but Trek technology again would easily cope. Whatever effect there was on tidal forces could be shrugged off by the human civilization, even if some other lifeforms here might complain.
If what exploded there was big enough to remove half the moon from existence altogether, one might expect there to be all sorts of exotic damage to nearby planets, though. No telling what, until we know more about the nature of the explosion.
Timo Saloniemi