Can a node exit the bitcoin network and return to his address and transactions?
Transactions are not directly sent to the node that holds the private keys for the addresses involving it. Rather transactions are broadcast to all nodes on the network and then recorded on the blockchain. When a node rejoins the network, it downloads and verifies the blockchain from its peers. When it does that, it scans the blockchain for transactions that pertain to it.

The process of receiving is completely passive; there is no direct sending to the receiver (the sender does not have that information) nor is there a claiming process.