In the manga Gymleaders get paid a wage by the Pokémon Company (or a similar name).
But I don't think they earn that money just by fighting other trainers. And it's probably not enough to live, seeing as a lot of gymleaders have side-jobs. Like Roxanne who teaches at the Trainer School and others mentioned earlier.
They're most likely paid to keep their gym in order, with little extra's.
I don't know about the E4, since it's only in the game that you have to battle them.
In the anime and manga, they're scattered throughout the regions... But they have obviously a lot of say in the bussiness of the Pokémon World. Seeing how Lance tries to tackle Team Rocket, Team Aqua and Team Magma.