The one major difference between WordPress.com and WordPress.org is the hosting. With WordPress.org, you host your own website. That means you are using an external hosting service like hostgator, godaddy, bluehost, etc..
No. With WordPress.org, you can use any domain extension: .com, .net, .co and others.
It will work on self-hosted WordPress.org or if you have a business plan on WordPress.com
No, it’s a child theme of Genesis and will need to have Genesis installed in order to work properly.
Yes, you can upload your own logo images, Preferably transparent images.
Yes! You can do that in Appearance > Customize > Colors or Fonts.
Yes, both content and widgets are included.