I love Mustangs, I have 2 at this moment. One of them is my main riding/using horse. IMHO, they are just like any other breed, with the proper training, they can do pretty much anything you want to. On the other hand, they are generally more of a 'jack of all trades, master of none' type horse. They can do pretty much anything in the lower levels but if you are looking to go far in some event, then you would likely have better luck with a horse bred for it.
This is Dobe, my main guy. I got him as an unhandled 3 year old stud and while it took a lot of time and work to get him to the point he's at now, I can do almost anything on him. He has a heart the size of Texas and will kill himself before he quits me. He's pulled cattle that weighed more than he did and tries his best to be a cutting horse though he isn't terribly quick on his feet. He is one of the smoothest traveling horses I have ever ridden and he's very smart and loyal.
This was just a couple of days after we brought him home. He was about as wild as a rabid coyote and almost as vicious. When I got him, he was this fugly little scrawny blown out thing that looked as though he could barely hold himself up, let alone carry a rider.
He was about 4 in this picture. He had put on about 100 pounds of weight when this picture was taken.
And this is what he looks like now after almost 6 years under saddle.
Then I have my other boy Koda. He's a nice enough horse but we just don't seem to click and one of these days I will give him a refresher and get him a new home. He was a 4 year old when we got him but since neither my Dad nor I had time, he got turned out to pasture until he was 5. That's when I decided to bring him up and break him to ride. He had been handled enough that he wasn't really scared of people but he still didn't know anything, not even how to lead. About an hour after getting him up out of the pasture, I was trotting circles on him in the round pen and he's never offered to buck. I haven't ridden him in a long time just because I don't have time.
This was the 3rd time he ever had a saddle on.
And this was after the last time I rode him.
Whether or not they would make a good family horse depends entirely on the individual personality of the horse. I really don't believe that Dobe will ever be kid safe. It's not that he isn't well trained or very broke, because he is. But he has a more reactive personality and is very quick to respond to even the slightest cue, but he also spooks sometimes. Koda would make a very nice family horse with a couple more years of riding and a few more million miles. He's got a nice temperament, but he is still what I would call green broke and isn't always entirely reliable because of that.
To answer your other questions, yes, they can show and be just as calm (or calmer) than any stock horse. I carry the flag on Dobe every year in the parade here in town. In my experience, they don't care what kind of horses they are with. Both of mine share a pasture with different horses ranging from a 34" tall mini-burro to a 17 hand mule to an 18 hand percheron and they get along great. Dobe is the herd alpha and can be rather aggressive if someone challenges him but Koda is right about in the middle of the pecking order.