1. Was the genesis block mined or just set somehow?
Both. The genesis block needed to be mined as it must have a valid proof of work. However it is not a block that the network mined but rather satoshi mined it and hard coded its value into the Bitcoin source code.
2. How did it all actually start? Did Satoshi share the first bitcoin software to a group of interested individuals?
Satoshi posted the Bitcoin whitepaper to the
Cryptography mailing list. He also registered the domain bitcoin.org (bitcoin.com was taken at the time) and hosted the whitepaper there. A few months later, Satoshi announced the first release of Bitcoin on the same cryptography mailing list. The source code and the binaries were hosted on bitcoin.org.
3. Did he announce a particular point in time to start mining? Or perhaps he had to mine by himself for a period of time before other people jumped in?
No. There was no start time to mining. Mining could be done by anyone who had the software, including Satoshi. The first block after the genesis block was mined later the same day, possibly by Satoshi. Satoshi certainly did mine by himself for some time, but other people were also mining. It is difficult to know which blocks were mined by Satoshi and which blocks were mined by someone else.
4. How long did they have to mine empty blocks?
The first transaction was block 170
5. With only a handful of nodes at the beginning, did they have to set a particular type of network communication between them?
The network protocol is largely the same as it is now. The same message types are used and the format is very similar. To find nodes to connect to, the original client would connect to an IRC server and post its IP address there. New nodes would connect to the same IRC server and retrieve IP addresses to connect to. There was no method to tell the software to initiate a connection with a specific IP address.
6. Did it ever happen that (during this initial phase) the network was off line for some time?
Yes. There were periods of time where blocks were not being mined (a few hours usually).
