I like the idea overall, but I'm afraid it might be too easy to abuse by ceaselessly spam-hunting to farm H/P.
Maybe hunting should yield H/P only during a special event akin to tournaments?
There's no problem at all with hunting increasing skills though, given you'll have to spend at least as much money/time as you do in training grounds.
Yes, require it to be a Hunting Party or whatever. Require a building Hunting Lodge to run them and a gold cost to perform them. Make them a social event just like tournaments.
