To increase the level of a Pokémon, it has to participate in a battle against a creature that gives experience (most creature, including non-Pokémon should give experience, you can find the exact amount given in the info tab of that creature in the Pokéworld section).

Once the creature is killed (by anyone / anything), the experience it can give will be shared to up to 8 Pokémon who participated in the battle (who dealt damage to the creature)

Does the exp. split among all of the pokemon?

Do you think it might be a good idea to make the pokeball belts act like ranged weapons specifially for hunting purposes? That way a trainer pawn can "hunt" using their pokemon as fighters and make sure that the exp goes to them. Maybe make it a toggle for gun hunting vs pokemon hunting?

To be clear, I don't think pokeballs should be thrown as the hunting weapon itself, just that the battling would benefit from becoming somewhat automated. Catching should not be automatic unless maybe there's a right click force option specific to catching so that you don't need to have drafted the pawn.

For example, a pawn is assigned to hunting and is given a pokeball belt and no ranged weapon. Said pawn approaches a pokemon the player has marked for hunting, and stands back while their assigned pokemon weaken the wild pokemon. The player then has the option to right click before the wild pokemon faints and select "catch pokemon." If the pokemon faints, the pawn slaughters as normal and moves on.