If he is a Dachshund cross then he will keep the longer back and shorter legs. They can suffer from spinal problems. He may well have trouble going for long walks and certainly you would have to be careful letting him jump on furniture or going upstairs, and even picking him up has to be done carefully.
Although it is a lovely thing to rescue a dog, I think that with all the problems you have had I would be more inclined to choose a breed and pick a puppy so you can see the adults and judge their temperament and type, and them let the puppy grow up to your ways, rather than take on something that you don't know the background/outcome of.
Crufts is on in a few weeks, why not visit there and go to Discover Dogs, where there are nearly 200 breeds on show, with people their willing to tell you all the good and bad points of their breeds, and you can see the dogs and make up your minds knowing exactly what they will turn out like.