Referring to the original TOS pilot pitch and the TNG Writer's Technical Manual, we can ascertain that Gene Roddenberry was adamant about two things in particular;
1) There ain't no money inside the Federation. None. All attempts to prove otherwise (usually by citing off-hand comments about 'earning your pay' from TOS) are thus doomed to failure. That provides quite a neat answer to your first question about how people are paid. They aren't.
2) Earth is a garden paradise. The vast majority of industry in the 22nd and 23rd century is done off-world and most everything in the 24th Century is done by replicator (obviating the need for heavy machinery and pollution). No-one needs to work if they don't want to, however the vast majority of people choose to work because it provides them with a sense of purpose and fulfillment. Ongoing education is also apparently ubiquitous.
There appears to be full employment and the existence of AI, robotics, transporters and "automated janitorial systems" likely makes menial work unnecessary.
In answer to your specific questions;
- How does society exchange productivity between its members?
They don't. There's no money. If you want something, you can simply replicate it. If you want something big, you queue up at an industrial replicator. If you want something rare or precious (such as land for a farm) presumably you can apply for it in a lottery or to a commission intended to apportion such things equitably. With off-world colonies siphoning off population, we can assume that those that remain are those that are happy with the present system.
- Is there any evidence that these enterprises still exist in some way, shape, or form in the 22nd, 23rd, and 24th centuries?
In TNG: The Neutral Zone, we learn the fate of money. It was negated, along with the banks responsible for holding it and tracking its movement.
PICARD: Here's what I propose. You can't stay on the Enterprise, but I have arranged for us to rendezvous with the USS Charleston,
bound for Earth. They will deliver you there.
RALPH: Then what will happen to us? There's no trace of my money. My office is gone. What will I do? How will I live?
PICARD: This is the twenty fourth century. Material needs no longer exist.
At the same time I think we can reasonably assume that for-profit corporations were also abolished although it's likely that some brands do still exist (Nokia, Chateau Picard, etc). Since these brands can't sell products (no money) and since there's nothing to stop me replicating the latest model, my assumption is that these would probably be managed as open-source projects with a controlling board.