The miners claim the reward themselves, but if they claim it wrong, the network will reject their block.  So in a sense it's the network as a whole that decides whether the miner has claimed the right reward.

The strength of the bitcoin network is having a larger number of nodes all running software which agrees on the rules of how the system works.  Changing the rules involves convincing as large a portion as possible of the network to agree to the changes, which makes the rules stable.

When you receive bitcoins into your private wallet, you ask the wallet to generate a private key and associated "address", and then you can transfer as much or as little as you like into that address.  Or generate multiple addresses and split things up.  It is up to you, regardless of the amount being stored.

Note that you never "receive" a private key - they should never be sent over the network, they are secret.