It's a lengthy and involved process, only two new breeds have been successfully added in, oh gosh, I think over 50 years. (Mini Satin and Thrianta.) The Lionhead has been trying to develop enough consensus and consistency for a number of years, and still don't seem to be getting close!
Here's the FAQ from ARBA, there are more detailed directions in the Standard of Perfection.
The definition of "Purebred" means two like parents that reproduce themselves 100% of the time. You would have to read the ARBA policies on accepting new breeds to see what they are open to and what doesn't have a snowball's chance in H E double hockey sticks of passing
PS Rachel you forgot the Mini Rex and the Jersey Woolie and the American Fuzzy Lop!also Florida Whites and Satin Angora