When I move an address to another account, and the address contains bitcoins, will the person who gets the address will get the bitcoins that where on the address?
Short answer is no. Accounts are a way to keep track of coins in "sub-wallets" of your wallet. When you receive coins to an address, the balance of that account is increased. After that, though, the link between those coins and the address balance stops.
Moving the address to a different account doesn't take all coins that the address has ever received with it, it only indicates that any future coins received on that address should go to the new account specified.
Similarly, moving the account balances doesn't change the amount received/owned on a particular address. Accounts are really just an accounting feature to keep sub-wallets, but moving money around in your accounts doesn't actually move them on the blockchain.