Is incus-compose a part of the base system, or an add-on
it's a program you install on your client machine. It just connects to the Incus API (probably using the Incus cli).
How does it work on IncusOS if it’s a command-line tool and there’s neither a remote nor a local shell, only a web API?
The Incus cli connects to the API.
Does it provide a built-in way to automate container updates
For the most part, Incus containers are system containers, similar to lightweight virtual machines. They run a full distro, including systemd and all that. For keeping those updating, you would want to use the auto update methods that Linux distros provide.
There are also OCI containers that Incus can run, but I don't think there is any form of auto update for those. You might be better off just running docker/podman/kubes in a VM for that.

Discourse is especially neat in that when you start typing a post, it automatically searches the title of your post and shows you recommendations. Here is an example from Incus' forums:
This is probably very, very useful and cuts down a lot on duplicate questions. But Matrix doesn't have anything like that, which probably causes more burdens on people answering questions.