Please have a look at this article that will provide you all the necessary information on how to set up your site to use an external hosting service provider - http://kb.worldsecuresystems.com/157/bc_157.html?bc-partner
Yes that is correct. The article provided above is only when you want to change the hosting service provider (DNS service provider).
In order to use all the Business Catalyst features (e.g. eCommerce, Web Apps etc.) you have to have the site created and stored on our platform.
You can take the content of web pages, templates (because it's HTML, CSS etc.) however you cannot take the way it works with BC. It's the same thing as taking the furniture out of a house. You can take it but the house you can't