Question:  Is there such thing as a non-registered (not papered) purebred dog?

Answer: Yes and No

Answer: Yes - One reason could be a person selling a unregistered dog doesn't register the puppies because they told their breeder that they only wanted a pet, therefore didn't want to buy the breeding rights on the dog (which costs more). When you choose the pet option you pay less and part of the contract will include the buyers commitment to spay or neuter the puppy at a later date. The breeder keeps the registration papers in their name until proof is provided by the buyer this has occurred.  The spay or neuter is considered part of the purchase agreement, just like the payment. If the buyer does not hold up their end of the agreement the purchase is not complete. The buyer does not own breeding rights. They are purebred however the buyer did not follow through with their part of the contract therefore, will never receive the registered papers.  A purebred should be registered with a legitimate registry.

Another reason may be it's not their dog ; it could be someone’s lost pet, or even stolen. 

Answer: No There is no such thing as a non-registered purebred dog.

Unregistered dogs are not considered purebreds as there is no proof. A dog cannot be purebred without papers proving they are true to the breed. A dog without papers has no proven lineage that they or who they came from is a purebred. When a person wants to sell a purebred puppy without papers, by placing the word purebred (without the dog being registered) in an ad allows them to think they can charge more for a dog. An unregistered dog should cost no more than an adoption fee from a rescue or shelter. If you are considering paying for a dog with no traceable history or lineage consider saving a dog instead. 

Question: Why buy a dog that is registered (with papers) vs non registered (without papers) 

Answer: In addition to the information provided in the question/answer above; you will know the full lineage of your puppy and it's ancestors. When you purchase a puppy, it should be registered, health tested, and include a health guarantee. There will be a contract that will state that if you cannot keep the puppy, at any point of the dog's life, the breeder will take the puppy back and ensure it's safety. It will also state what the dog is intended for (show, working, pet or breeding rights).

Breeding is not fully regulated as anyone, anywhere can breed dogs as purebreds.

Question: What is difference between a pocket and standard bully?

Answer:  Their height

The "pocket" type is a smaller variant, with full-grown males under 17 inches (43 cm), but no less than 14 inches (36 cm), at the withers and females under 16 inches (40 cm), but no less than 13 inches (33 cm), at the withers.

The standard American Bully type is a medium-sized dog with a compact bulky muscular body, heavy bone structure and blocky head. Male dogs must be 17 to 20 in (43 to 51 cm), while females must be 16 to 19 in (41 to 48 cm).