In the pools they often refer to something called a "share". Is there an exact correlation to the amount of work (i.e. hashes) done?

Aha so it also serves as proof that work is being done by the miner?

clever..

So the 'F':s in the middle also guarantees that it holds no bitcoin value?

When a bitcoin miner is at work, it tries many nonces x, such that sha256(sha256(block_header_with_nonce(x))) results in a number that is small enough.
Currently, this number needs to be smaller than

  000000000000DC31000000000000000000000000000000000000000000000000

(written in hex) for it to result in a real bitcoin block with value. Pools, however, (at least the ones i know about), only request hashes that are smaller than

  00000000FFFF0000000000000000000000000000000000000000000000000000

So, it will see all "real" hashes calculated by the miner, but also many hashes that are somewhat larger than this.
The pool gets its income from those real hashes calculated by the miners, and distributes them (using some formula)
according to the number of low-difficulty hashes it received from each miner.

No, you need to decide who (what address/public key) will earn the generated 50 BTC before starting to mine, since the hash depends on it. So once you find a real hash, you only have two options: broadcast it, and let the pool gain money (of which you'll get a piece), or discard it (which is bad for everyone).