are there dogs that catch mice and rats and do they do it better then cats

Pwca

I think I've read someplace (I'd have to find the book again) that Yorkies were mostly the pet-ized versions of working terriers in their originating area. Keep in mind though that Yorkies are supposed to be 4-6 pounds, and that's shrunk in the last hundred years- 10-12 pounds isn't too small to work, especially if the quarry is mice, rather than rats and groundhogs.

 I suspect yorkies were originally just a slightly functional but mostly companion dog- a lot like the smaller sizes of spitz and poodle.
 

I said the very small, most earthworking Jrts weigh about 12 -15 lbs, I know several that come in under that are very successful  working dogs to groundhog, fox, coon etc.

I meant terriers generally 10 lbs or less, now having said that I have a bitch that comes in at 9 lbs, she has successfully hunted in the earth to groundhog, but the risks are greater and she can and does lose the quarry because she can't hold them there, until we can get a draw dog to the earth.