Not exactly. Desktop environments (or whatever the program is that launches the application) decides to use systemd to run each application in its own cgroup.

Systemd is certainly relevant to the Linux desktop as a whole, especially regarding logind. But there's no specific relation to GUI desktop applications that I'm aware of at least.

No it doesn't. It could, but as far as I know this is not used at all.

There are user services, but that's a separate concept.