Well, you can’t get updates from other instances without it being connected to the internet and reachable. So for your usecase you do need it connected to the internet. On the LAN you will only be able to see other instances on the same LAN.

This could mean that multiple lemmy processes can share that folder but it would be a stretch to assume that a single process can serve multiple domains.

You need to run one Lemmy per domain then. I would like to see the parts of the doc that suggest that this is supported.

What do you mean by site? Having one process serve multiple instances with different domains? Or do you just want to define custom styling per community?