The doxie is going to have a million times more energy and playfullness than the cats. My doxie has never given up on trying to get the cats to chase around with him, at most they have started to tolerate him. We run with him and he runs and plays with his stuffed animals every evening when the cats ignore him. He visits my daughters ranch and runs with her little dogs. I think he does about 25mph. I'd get 2 of them if I could. They have a lot of energy.
I have a female dachshund and two male cats, they get along pretty good. my oldest cat is 8 he just tolerates the dachshund - she just turned a year old. My younger cat is 4 and they wrestle and chase each other everywhere. It depends on their personalities I think, try bringing a dog you know is good with cats into your home to see how the cats handle it. Dachshunds generally get along with cats there are always exceptions though.
Some will, some won't. They are hunting dogs and will chase anything that runs. Most rescues and shelters will be able to tell you if a dog will get a long with other types of pets, but you always need to introduce them slowly regardless.