How does Apple's two year warranty in Europe work?
It isn't quite correct that "the EU" has forced Apple to abide to these regulations. Rather, an Italian judge forced Apple to abide by Italian law. As a result, Apple has seen the light and adjusted its policies to comply with EU directives across the EU (as it was required to). 

EU countries implement EU directives, but generally have some freedom in those implementations. So details in individual countries may differ, although your questions are general enough that the answers below probably apply everywhere.

The regulations that Apple now has to comply with don't provide for warranty per-se. They do make retailers responsible for providing a product that is functional for its expected, reasonable lifetime.

In general:


Yes, the online shop is a retailer, so it is responsible for providing you with a functional product.
No, these regulations apply in the entire EU, and AFAIK all EU countries have implemented them. Apple must comply in the entire EU. However, details of the rules that Apple must comply with may differ in specific countries.
Not really. While Apple serves as a manufacturer and a retailer here, the EU rules specifically apply to retailers, not manufacturers. In addition, AppleCare may provide protections or services that retailers aren't generally expected to provide, or can't (as in your question 4.)
Maybe. Apple may provide this as a service, but it isn't required to. If you bought an Apple item from an EU Apple Store, you can't then require a US Apple store to service your device; it's a different retailer.