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Wow, I think giving bosses/big monsters multiple pools of HP is probably one of the most inspired ideas I've heard in a long time. My players got to the final fight of The Lost Star a couple weeks ago, and basically just rolled over on the final boss once they got in close with a good crit (the random giant rat in the fight managed to tank them in a hallway for like four rounds, it was hilarious), despite there being only one real damage dealer in the party (rogue).
I might also suggest coupling this idea with a "revenge" mechanic on certain fights - once one pool of HP is expended, the monster jumps initiative to lash out at the party with a special attack/spell/something - since depending on how initiative works out, the party might still roflstomp the boss despite the HP separation. I would definitely not suggest this be used for every monster/fight with a pooled HP system, since it could get pretty dickish very quickly.
I also really like the training level system proposed. Advantage/disadvantage in 5e is one mechanic I think more designers should steal and incorporate into their systems. I think your suggestion gets a little wonky with weapon attacks though - high level fighters suddenly roll 4d20 for one attack, which seems a little unnecessary. Maybe combine with the "eliminate the +1, -1..." suggestion:
- Untrained: roll 2d20, take the lower
- Trained: roll 1d20
- Expert: roll 2d20, take the better
- Master: roll 2d20, take the better, +1d4
- Legendary: roll 2d20, take the better, +1d8
Something like that perhaps.