Apple care

I assume you got the Asurion coverage from your cell provider? It is not something Apple offers. So, getting a replacement from Asurion will likely not qualify to purchase Applecare for it. In any case you would most likely have to check with your cell provider to make sure.

Not to my knowledge, since to my knowledge Asurion devices are not new, they are refurbished devices. You can double-check with them, or can use this to check with AppleCare. AppleCare Products - Apple

Rosafig Said:

"Apple care: If I receive a replacement device from Asurion can I add the new device to AppleCare?"

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Read the Asurion Pamphlet:

Go to a retailer that offers Asurion, and request a pamphlet pertaining to the service plan you purchased for your device. See what it reads in the find print.

no, but you can cancel the Applecare on the old device, get a partial refund, and purchase it for the new one (if it is new and eligible.).