can the new rechargeable mouse be used with existing mac air pc's from 2012?

If by "rechargeable mouse" you mean the Magic Mouse 2, then Yes, but the Mac needs to be running OSX 10.11 El Capitan or later:

http://www.apple.com/shop/product/MLA02LL/A/magic-mouse-2

System Requirements

Bluetooth-enabled Mac computer with OS X v10.11 or later

To a degree. The key words are on the box; to be used with El Capitan and above. That means you would need to upgrade from the Mavericks your profile says you are using. Once that is done, yes.