Thank you for this.
In general: Am I right that the nature of sending bitcoins is something like a stub of a balance. Either my sent bitcoins arrive at the new adress within 2-3 hours or the never will arrive at this adress.
You're actually asking questions about the technical part of bitcoin.
<simplified technical explanation>Bitcoin is a decentral ledger containing all confirmed transactions ever made. Each transaction is basically a bunch of scripts.
On one side of the transaction, you have the signed scripts used as an input of the transaction. The input scripts are signed with the private key belonging to the public key, belonging to the address that was used to send BTC to (to keep it simple.. In reality i'm skipping a couple of things).
On the other side of the transaction, you have scripts stipulating who can spend the output of the transaction. You basically say: the person who can provide a valid signature that can be verified using the public key from witch address [address] was derived can use this output as an input to create a new transaction.
After you generate this transaction, you broadcast it to the network. If the transaction is valid (the signatures are ok, the inputs are not yet used in a different transaction,...) it is accepted in the mempool of the nodes you broadcasted the transaction to. These nodes will broadcast your transaction to the nodes they are connected to, and so on, untill the full network knows about your unconfirmed transaction.
</simplified technical explanation>As long as your transaction remains in the mempool of a single mining node, it has a chance of being added to the block that particular miner is currently solving. If the miner finds a block below the target diff, and he's able to push the block to the network fast enough (so it becomes part of the main chain) your transaction is confirmed.
There is no 2-hour limit. As long as your transaction is floating around in mempools, it has a chance of being confirmed...