How does pools work?

As far as I see, the basic idea of mining, is that we get piece of data, where we need to try all variants of 4 specific bytes = 2^32 variants, we hash this data using SHA256 and if we get value meeting our difficulty requirements we "win"(?)

I wonder how does this work on lower level:

1) How it's made that different clients does not work on the same variants?
2) How it is prevented that client who found solution would "steal" 50BTC, instead of submitting solution to the pool server?
3) How does pool software process this solution to get 50BTC and notify whole system that everyone (including other pools & individuals) need to work on next block?
4) Is that correct that all fees associated with newly generated solution are "payed" to the owner of this solution in the nearest minutes after it's generation?

Sorry for newbie questions :-)