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After mining the genesis block, did Satochi have to leave his computer running to mine the subsequente empty blocks?
Was he running the whole network on a single node or did he require multiple nodes to test the transaction verifications?
What would happen if at some point all nodes were off and no transactions or blocks were being generated?
At some point the whole net work were probably a couple of personal computers. Did they run 24/7?
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After mining the genesis block, did Satochi have to leave his computer running to mine the subsequente empty blocks?
He didn't have to, and it appears that he actually didn't. Block #1 has a timestamp that's more than 5 days later than the genesis block. Any decent computer in 2009 would have been able to mine a block at difficulty 1 much faster than that, most likely within a couple of hours. (I have a computer I bought in 2010, not particularly fast by the standards of those time, and it produces about 2 MHash/sec, so that it would mine a difficulty 1 block in an average of about 2000 seconds or about 33 minutes.) So almost certainly, neither Satoshi nor anybody else was mining during those 5 days.
There's another gap of 24 hours between blocks #14 and #15, which suggests nobody was mining during that time period either.
Was he running the whole network on a single node or did he require multiple nodes to test the transaction verifications?
I don't think we know this. It's likely he did some sort of testing with multiple nodes prior to release, but I don't know of any data for how many nodes there were on the live network in the earliest days after release. We also would have no way to know for sure whether they were being operated by Satoshi or someone else.
What would happen if at some point all nodes were off and no transactions or blocks were being generated?
Then no blocks get generated, that's all. Nobody collects any block rewards if nobody is mining. If any transactions were created during that time, they could not have been confirmed until after someone started mining again. It's not inherently necessary that people be mining at all times - things start right back up when they come back. The software can't really tell the difference between "nobody is mining" and "people are mining, but due to bad luck there hasn't been a new block for a while".
At some point the whole net work were probably a couple of personal computers. Did they run 24/7?
As mentioned above, probably not.
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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.
Did you measure the gap of 24 hours between blocks #14 and #15 manually? What us the best way to explore the early blocks of the blockchain? – Pedro Gonçalves – 2018-01-01T02:06:13.977
Every block has a timestamp, which you can view with your favorite block explorer (e.g. go to http://blockchain.info and type "14" in the search box).
– Nate Eldredge – 2018-01-01T02:12:29.807