It sounds like somewhere down the line a chihuahua was bred with a terrier....the wire haired might have come from a mixed chi.  They might have looked mostly like chi. But being mixed with terrier in my opinion.....  Dichihuahua

The smaller the dog is ( Chihuahua ) the more compact his/her body is which includes their snout...which makes it harder for them to breathe so a lot of times they snort or gag.... Dichihuahua

I wouldn't for the puppies & his safety....most mom dogs will protect their babies....even from the dad   dichihuahua