Suppose your rat terrier was born before the AKC approved the breed. Is it now eligible to receive papers, even years after its birth? My 6 year-old rat terrier was a rescue so obviously she can't get papers in any case -- but I'm curious to know if this possible if the dog was bought from a breeder before they were AKC-recognized.  How does that work?

My rattie is AKC registered. I registered for the purposes of being able to compete in AKC sanctioned events like agility and earthdog. She happens to have a 6 generation pedigree from UKC so the process was simple. You can get any rat terrier registered with AKC, even the rescue. You do that by getting an ILP#, if you don't have a pedigree, or can't find the pedigree, or whatever.
Here's the info for getting an ILP#:
www.akc.org/reg/ilpex.cfm
It also says what akc events the dog may participate in with an ILP#. Rat Terriers are NOT eligible to participate in conformation in AKC just yet.

Also, the answer to your original question is yes. Any rat terrier, regardless of age can be registered or ILP'd with AKC. Mine was born before they were added.