Its possible you could, with alot of hard work over many generations, but any pups born would suffer from many problems, caused by the clashing of genes and DNA.
Yes, you can cross breed chickens and most of the time, we can't stop ours from cross breeding anyway as they all run around as a flock. The results mean they will not sell for very much but some can...
There are many breeds of dogs. There are probably over 250. If you go to the Dog Breeds link below they have all the breeds listed A-Z with pictures and information on that particular breed of dog....