Do mix breed dogs have less medical problems?  

Genes come in pairs, one from mom and one from dad. Many gene are what we call recessive. A recessive gene can be hidden by a dominant one. So if an animal gets a defective gene from mom, often the one from dad can pick up the slack and mask the defect. But if it gets the defective one from both the defect will show up in the individual. (This is over simplified and not all genes are on a strict dominant/recessive relationship, but it gets us started)

When we create a breeding pattern to select for certain features of an animal (short legs, good sense of smell, long ears, speed, or whatever) We stand a good chance of accidentally also selecting for other genes. We also limit the gene pool so these genes are more likely to show up in the individuals. This can be seen in Dalmations who are prone to epilepsy, German Shepards who are prone to hip problems, and many others.

There is also the issue as in the case of bassette hounds, Shitzu, pugs and others where the feature that is artificially selected for is detrimental for not genetic reasons. For the "long dogs" they will have back problems becuase of the length of the spine and the sag it will get in the middle. For "short nose" dogs, they will have sinus and breathing problems because their noses are the way they are.