We have a rescue dog (mixed breed, 70lbs), now 7 years old. He is great with all our indoor birds and rabbits. We did take him to pretty intense training from a professional trainer (she trained police dogs formerly, and specializes in bully breeds) when he was a puppy, but I also think some is his temperament (likely combined with earlier training). Training teaches the dog what to expect from you, and who the alpha/boss is. I would recommend extreme caution with a dog around other pets for quite awhile.
I should also say our dog does not have a dominant personality which we were able to tell when he was just 3 months old. When we picked him out, he was calmer, but still made good eye contact. He was a bit skittish, but again training really helps with that. I can not emphasis the value of training a dog, especially as a puppy, before they develop bad habits. Our dog is so well behaved now, people even comment on his good behavior. It also just makes him an absolute joy, it takes the stress of having an unruly dog away. He is just fantastic.