The custom code FAQ linked to above should answer most of your questions, but with regard to this one...
CORE will actually run in the SmartThings cloud, on the company's servers.
You will do a one time setup to publish it to your account by using the IDE in a web browser.
Once it's installed in your account, you'll be able to set up individual rules, called Pistons, by using the SmartThings mobile app on your phone. But when the rules actually run, they'll be running in the cloud.
You can see piston examples in the peer assistance thread: