1. You'll have to check with the hosting service on that. Transfer processes are different for many of them.
2. I recommend that you transfer both the domain and old site as-is over to the new host, that way there's no interruption in the site.
Install and build the WordPress site on a subdomain, like new.hatsandmittens.org (which you'll make through the hosting provider's control panel).
Once you're done with the WordPress site, move it over to the main domain to replace the old site following this guide: https://codex.wordpress.org/Moving_WordPress#Changing_Your_Domain_Name_and_URLs