Would this node inmediately continue proof of work even if the transaction pool is empty?
Yes. The block doesn't have to include any other transactions. Some blocks only contain the Coinbase.
What would happen if he mine the block and new transactions would come in? He would to start again with the new transaction because the tx are included in the block so the hash of the block changes, so he each time new tx are added the node has to restart the PoW, Right?
No, he doesn't have to. The transactions are already technically included in the block
before the miner starts hashing the header. The header contains the merkle roots which is the hash of all the transactions that the miner wishes to include. Unless the miners wants to waste his resources, they won't discard a
valid block which fulfills the target even if new transactions with larger fees gets relayed to them.
There is no restrictions about how many transactions a miner should include. He is free to decide. [Read DannyHamilton's part on this.]