About AppleCare +

The period during which you can buy it is 60 days in the US. And, since it is automatically assigned to your Apple ID and device during purchase, I doubt very seriously that someone can buy it for you - you best check with Apple on that. In the "old days", you could actually buy a small box which contained the serial registration code, so anyone could buy that and give it to you as a gift - that method has not worked for some time now. Everything is done electronically now and registered automatically. Here are Apple's instructions for various products:

https://www.apple.com/support/products/

Hi,

Yes, you can.

Hi,

You can purchase AppleCare+ within 90 days(it depends on your country, try to check your country AppleCare site) from you purchased your MacBook Air.

Please remember that it is not a good idea to give someone your Apple ID and password. They would be able to access your account and spend your money.