Do people run root GUIs as a client? That seems silly to me. I don't have one single GUI program that I run as root unless I'm troubleshooting a permissions issue.
Even if you trust the programs you run, they can have their own unintended vulnerabilities.
No, but that's where escalation comes in. You go to a page which uses javascript to take over your browser. Now your browser can capture and send back your shell password captured from the terminal window.
When you need to su or sudo, do you switch away from X to a separate virtual console, or do you just do it in an xterm (or equivalent)?
How about when you want to ssh to a server and do something, possibly including su or sudo on the server. Again, do you do that from an xterm or equivalent on your desktop machine, or do you switch out to a separate virtual console for all your ssh activity?