After setting the CatalogURL on the server — just like how the CatalogURL has been set on the clients — the server can self-update from the local stash of updates, yes. The server can be set to update from Apple — the default, and the way your system has beem configured — or potentially from a different software update server, too.
You can but the default is to update from apple
It's a bit of a waste of time having the server sus updating from itself unless you have multiple servers requiring updates
Self-updating is one less download from Apple, FWIW... If you're on a larger network link, that's less of an issue.
Letting the server download updates from Apple rather than the SUS can also be less downloads from apple
It depends on how you mange the SUS - don't have SUS download server updates, and any other updates that aren't required.
I can't see the point of having a single server self updating from its own SUS
More hassle than what it's worth IMO, I prefer to have the server check for updates from Apple download and install
Then manually have to find out about updates then have the SUS mirror the correct release then run software update on the server and update
But each to his own