If you get a puppy, the cat and a basenji should be fine. An adult B meeting a cat for the first time will typically give chase. Cats run from dogs, typically, and basenji's chase anything that runs away, including people. 
Dusty is darling1 How great that you ignored the trainer and figured out your little guy. They can be so smart that they "fake it" in many ways, making it hard to realize they are deaf, or don't see well. Kudos to you for being such a great mom!
Thank you, I think he's a doll myself but I'm partial. You know with him not being able to hear he is very visual and he will watch TV with me. I find that so interesting. He likes animated movies like the ones from Pixar, I can not watch any program with dogs, cats, horses, cows, bears or basically any animal because he starts barking like he is going to be attacked at any given moment. But if we are outside, and we see dogs or other animals he doesn't do anything. Maybe he is just protecting his turf when he's inside. When I tell people that he is deaf, they always say 'oh poor thing' and I always tell them that he doesn't know anything different so he's fine. If he had been able to hear and the lost his hearing that would be much harder for him. But he's really smart and picked up on my sign language very quickly, so he's happy. I'm happy too. Although when he's being a bit unruly I find myself scolding him then I laugh because he can't hear me. Then I tell him 'lucky for you that you can't hear me mister' and I laugh.
