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. 
Hi Deborah. I used to live in Bucks County. You ever need a new vet, try Pets Best Friends. My friend Brenda works there as a vet tech and might be able to help you find a positive trainer.
Sometimes very mouthy dogs can be taught to have soft mouths and learn no teeth, but it does seem for some the putting of human parts in mouth is almost irresistible.
Kathy Davis' article for no bite works, but forget the yelping advice… many B's consider that the equivalent of a squeak toy and do it more.
http://www.veterinarypartner.com/Content.plx?P=A&C=23&A=1128&S=0&EVetID=3001644
For lessons in all obedience, but a truly positive trainer, try this site.
http://www.clickerlessons.com/index.htm
Mary is down in Delaware, but you can contact her also for advice. She is superb.
