Gym Leaders would probably make money how other trainers do. By taking half of the loser's money. They probably have to pay a license fee to the Pokemon League. Labs could be funded in many ways. Maybe the Pokemon League or some rich organisation funds them for research. Or maybe professors are just retired trainers that made all their money when they were trainers.
The Elite Four is probably the same as Gym Leaders, only they'd probably be paid also, rather than have to pay a license fee.
The Elite Four is probably the same as Gym Leaders, only they'd probably be paid also, rather than have to pay a license fee.