Buying the apple care in US, am I covered in case I move to another country?

May I suggest that you check with Apple to be absolutely sure as the legal language in the documents is a bit vague. Here is an excerpt:

Please note that if you seek service under this Plan in a country other than your country of purchase, the service fee will need to be paid in that country’s currency and at that country’s applicable rate – for further details, including regarding applicable fees by country, please visit the AppleCare+ support website at apple.com/legal/sales-support/applecare/applecareplus/

That would seem to indicate that it is covered. Later on, it states:

Apple may restrict Hardware Service and ADH Services to the country where the Covered Equipment was originally purchased.

Note the rather ambiguous term “may restrict”.

All of it can be found here:

https://www.apple.com/legal/sales-support/applecare/applecareplus/1020/101320_applecareplus_us.pdf

The regular 1 year warranty includes the same verbiage (“may restrict”):

https://www.apple.com/legal/warranty/products/ios-warranty-document-us.html

No.

AppleCare and AppleCare+, like the standard warranty, are only valid in the country where the phone was originally sold.