After mining the genesis block, did Satochi have to leave his computer running to mine the subsequente empty blocks?
You can be sure that he had to leave his computer for a while because the hash of the genesis block did already start with 10 zeros. That's a lot for an average computer.
Was he running the whole network on a single node or did he require multiple nodes to test the transaction verifications?
I think, it's obvious that he wanted to test his own idea in a "almost real scenario" and so I think that he used more than 1 node.
What would happen if at some point all nodes were off and no transactions or blocks were being generated?
Nothing happens. The nodes do this have the blockchain(s) in their memory. That Satoshi could not make transactions but the transaction history was still there.
At some point the whole net work were probably a couple of personal computers. Did they run 24/7?
Probably a couple of them did do that to experiment.
But: We don't even know who Satoshi Nakamoto is. So we do also not know how he did it exactly.