Ok I'll give your questions a shot!
1. who registers the puppy you or the breeder?
You both do, the Breeder send in his/her paper work, and you send in your part. The new name of the puppy & naming you the owner. Yes, they charge you.
2. what does it mean to have papers? (meaning what are they used for ?)
Papers are used for Breeding, really just to prove you have a Yorkie.
3. how does papers affect prices?
Most Breeders will charge more if papered, They have to pay for the papers too, that might be why.
4. whats the backlash for not registering
I don't think there is one. As you would be the only one who knows you didn't register. You don't ever use the number unless you are Showing or Breeding.
Now, I am not a Breeder, and I don't Show my baby, So these answers, are only my guess. I'm sure someone else will post. I have three babies, Princess & Mandy are papered, Poncho is not. I love them all the same!