I wonder how does this work on lower level:
Me too. Lets try to help each other. Not sure if all my answers are correct, but I will try...
1) How it's made that different clients does not work on the same variants?
I think this depends on the pool server, handing out different shares to the miners. Maybe you get some info on this at the mining part of the forum or at some good pool.
2) How it is prevented that client who found solution would "steal" 50BTC, instead of submitting solution to the pool server?
To the extent I understand this: You only need an already prehashed part of the block to mine, but the entire block to claim the bounty. Pool server has the block; miner only has the prehashed part. This might be wrong; I am still trying to understand this myself.
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?
The pool software gets the value of the nonce and then can complete the block, relaying it to all other nodes.
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?
Payed? Actually by solving the block the block solver becomes already owner of the transaction which grants the money to him. It will take a while until other blocks are found and the chain becomes longer - and this can be trusted.