I don’t want to go on for days and days about hip dysplasia, although I could, but no, there’s no real data that would indicate that it’s less common in mixed breeds. It tends to follow a trend line of heavier bodies, regardless of size. So Pugs have an extraordinarily high rate of dysplasia, as do the heavy medium-sized dogs (Bassets, Bulldogs), the heavy large dogs (Clumbers, Labs) and the heavy giant dogs (mastiffs, bloodhounds). The lighter, “racier” dogs have a much lower rate. Greyhounds have virtually none at all, no matter how randomly they’re bred.
What this means in terms of mixed breeds is that if you’ve got a big heavy mix, it’s got a very good chance of having iffy hips, because it’s big and heavy AND because it doesn’t have any health testing behind it. There’s no reason to think that a Lab/Bloodhound/St. Bernard mix is going to have any better hips than Labs, Bloodhounds, or Saints. Ditto a Lhasa/Basset/Cocker, or even a Pekingese/pug.
The best way to protect yourself against hip diseases, if you want one of the larger or heavier dogs, is actually to buy from an excellent purebred breeder who has generations of hip testing behind the dogs (so has managed to buck the trend and produce good hips on his or her dogs) and offers some kind of health warranty so you get a replacement puppy if yours turns out crippled.
Many people would rather adopt, which is wonderful. In that case, when you are looking at rescuing a large-breed puppy, you need to feed super carefully (Never Puppy Food!), keep the dog very slim when he or she is growing, and encourage lots and lots of free exercise (long walks, hikes, playing, etc.).
If you really want a dysplastic dog, go buy a large-breed purebred or first-generation mix (“designer dog”) from a careless breeder with no health testing, make sure you feed puppy food and maintain a nice rolly-polly puppy, and keep the dog crated a lot.
Oh, and you can do Penn Hip on puppies as young as 16 weeks. It’s a lot of money to throw at a problem that may or may not exist, but if you need the information it’s certainly possible to get a decent hip picture that early. I would discourage you from doing any other method at that age, though–some vets say they can diagnose via palpation; others will look at traditional OFA-style films. Neither is anywhere close to accurate at that age. Penn Hip when young, OFA is fine for 18 months and above.
Ruffly Speaking: Railing against idiocy since 2004 