The most capable, adaptable, and accomplished adults I know moved around a bunch as kids. It forces you to learn how to live in the world and get out of your comfort zone. The most anxious, bitter, and underperforming adults I know grew up in the prototypical suburban bubble environment. End meaningless personal anecdote.
I still agree, and in fact kids are often more adaptable than adults. Usually they can handle change, and it is good for the mind to experience different worlds.
Depends on the country / school system. Not all kids have 3 months of "summer break."
I haven't lived in one place longer than two years since I was eight or so. It wasn't the easiest thing in the world, but I wouldn't be the person I am today if I hadn't grown up that way.
I guess there's a happy medium. I've heard horror stories from both ends of the stability spectrum.