Yes they usally do but if you make sure that it does not do any high jumping then it should be fine.
Back problems are common, but no more so for the miniature than then standard size. The other medical problem that can show up is seizures. Miniatures usually have one or the other of these issues. (Mine has escaped the back issues, but has OCCASIONAL seizures (about one or less a year)
My doxie had surgery last year for a disc and 10 month later we caught it early enough that so far he is only on meds and seems to be responding well to them. Not to scare yu but if the doxie needs back surgery its a pretty penny...$5000 is what came out of my pocket for it.
Back problems do happen, so you need to take caution with them running up and down stairs and jumping up onto furniture. I have a mini as well as a full sized and i have always tried to teach them not to jump but they always want to jump up to get into the bed! Doxies are usually little daredevils so luckily i have not run into any back prbblems with them so far.