Yes, you can. It's a relatively recent addition. The original commenter probably doesn't see much difference between copilot and Cursor because copilot has caught up quite a lot in the last couple months
I'm not aware of that, but I also lost access to Copilot when we switched to Cursor so I can't check at the moment. That's great if you can because Claude is why people like Cursor.
For ask/edit models: yes, but you have to enable them (which you don't get a choice over if it's an account managed by your company -- just saying)
For completion models: no, they retired gpt-35-codex, leaving only 4o-copilot.
From my experience (GH Copilot Pro with engaged_oss SKU, ie. for free), after they announced their "pricing changes", they also made the performance worse. The completions went from good, to actively distracting. tldr; they enshittified it / did a rugpull.
Right now, it is not worth paying for. ChatGPT Plus is far better value for money if you don't care about autocompletions/pure vibe coding.