1. Crosstraining happens much less than people seem to think. In my experience, dwarfs will only attend weapon skill lessons if they already have experience (can be dabbling) in the skill. A speardwarf who never touched an axe will not receive axe lessons from an axedwarf in the same squad.
I've seen cross-training to the tune of maybe a skill level per year when i "contaminated" the skill pool by having dwarfs switch weapons during their career; otherwise, cross-training just doesn't happen. It's a minor issue, and if you directly issue their "for-keeps" weapon types to your recruits, it doesn't happen in the first place. Warning: once crossover experience is present, the dwarf is forever eligible for demonstration training in that skill. Demonstrations check experience/skills, not equipment.
2. Mining quite likely just isn't available for demonstrations: i've only ever had one war miner; she only has miner skill, no standard weapon skills, and she never organised mining demonstrations. There's a dwarf in the squad who immigrated as miner (and still has mining active as labour), and he's evidently never received a refresher mining lesson from the legendary war-miner, he's listed as "dabbling".
PS: Just to drive home the point that cross-training is not a concern normally: the squad with the war-miner is a massively mixed one using all dwarf-made melee weapons - axe, spear, sword, hammer, mace, pick. The only cross-skill weapon knowledge in the entire squad are the "dabbling" miner skill of the macedwarf (immigrated with and later decayed through negligence) and an "adequate (rusty)" sword skill of a speardwarf (must have touched a sword sometime; clearly hasn't seen much cross-training either, or the skill wouldn't have gone rusty). The other eight - all legendary, trained together for years - have no crossover weapon skill whatsoever.