AppleCare on the prizes from the new iPad contest?
Apple will register a device to anyone who presents it for service. It doesn't mean some database doesn't still remember to whom it was sold or registered in the past, but in practice it doesn't matter since possession is all that matters.

As to paying for AppleCare, you should get it in the country where you live if at all possible. That ensures you don't have to ship the device elsewhere if asked to do so. The plans are customized for the laws of each country - and in some cases, customized for states or provinces as needed within a country.

Assuming the device was purchased from the US - you should also be fine buying AppleCare from the US store. Again, the warranty and service are valid worldwide, but in some cases, you will be responsible for shipping it to the country of purchase / AppleCare in cases where local service is not readily available.