Matrix won’t necessarily download all state/messages automatically, but if your client requests a non-available message your matrix server will query other matrix servers for it (backfilling).

E.g. if you scroll up to older messages, it might take a a few seconds but your client should eventually show them.

Matrix server use a back-off for servers sending messages, so if your server is offline for many hours, it might take a day for your servers to get messages pushed to by other servers again.

Took a look at the specification, this is what I found:

For federated servers performing delivery to a third party server, delivery SHOULD be performed asynchronously, and SHOULD additionally retry delivery to recipients if it fails due to network error.

So they should retry. Note that should is not the same as must. So there is no obligation. There is no timeline in the spec about for how long or how often retries should be done. The wording says network error.

My interpretation: the spec leaves a lot of room for implementations to differ. Network problems don’t normally last for days though. I’d guess that if your server is down for 5 minutes, you’ll still receive most or everything you’d normally receive. I wouldn’t trust on that if your server is offline for more than a day.

No. There’s nothing for people to connect to when it’s shutdown. And requests or submissions would just get bounced back by the server not existing.