What I like to do is go to the store and mime how I'd like my door to swing. It works.

I'd use the underlay made for tile carpets. There is a specific kind.

An underlay is a good idea. It will help with floor temperature and sound. It will help to protect the subfloor from spills and damage.

You as far as fixing them, I wouldn't unless they're moving around. You could use carpet tape to help tiles stay in place.