The only files that you need to move are the files form your theme, plugins and any uploaded files, so pretty much jsut the contents of the /wp-content/ folder. After that transfer the database and that’s it.
Jetpack is just a plugin so that should work the same way, but I don’t use it so I can’t say 100% sure. It should though.
I don’t do free themes. The couple that I’ve found were just terrible. Bad coding was just the start. I’ve found that msot of them set up links back to the free template site without asking, and two of them actively cloaked links to other 3rd party sites that I did not want to be associated with (gambling, pharma and more…) so I will never use a free theme again. I don’t even use commercial themes. They are normally to bloated with features that no one will ever want, but look cool and give it another dot-point for it’s features list. I develop all of my own themes. I do use the base files from the default WP themes as a starting point, but that’s only to get the page structure right, after that everything else is custom.