How common is it to receive letter from admissions asking you to apply, out of the blue?

Its quite common. I remember being excited that Brown started emailing my daughter in her sophomore year after her PSAT testing. Its an automated system that takes scores they like and sorts out kids that might make the cut. Also if your kid listed certain schools that they are interested in while registering, those schools will start contacting you to apply.
As @IxnayBob said- it wont stop once they head off to their selected school either. Hofstra is convinced that my daughter should leave Yale and transfer to them. They say so at least 3 times a week. LOL