can you bring in a carry on luggage into theatre. leaving for airport right after?

You should call the theater to ask what arrangements they have.

Most theaters allow you to check your bags.

Definitely not by your feet. If you're over 5'0", there's barely room for your legs! There is a coat check, I believe, though.

I am not sure as the seating area are very compact but if you ask the staff they maybe able to keep it for you