Check what a site does on plain HTTP: whether it answers at all, and whether it sends visitors straight to HTTPS the way it should.
Port 80 should exist only to move people to port 443. Your mail records are a separate question and a more consequential one.
Whether a site responds on plain HTTP, the status code, and whether it redirects to HTTPS.
Yes. A permanent redirect from http to https is standard practice for security and SEO.