If you want to be nitpicky and technical, you certainly can and may purchase AppleCare after that one-year period, you just can't register it to the product w/ the expired coverage.
Keep in mind though, your phone support only lasts 90 days, but will kick in once you buy AppleCare. So, after that 90 days, if you need phone support (not related to hardware) you'll need AppleCare.
Yes, but you do not have to purchase - AND REGISTER THE APP - until the one-year point minus one day (364th day of registered ownership).
The difference (and ONLY difference, AFAIK), is that if you purchase immediately, you get free phone support after the initial 90-day period has expired.