Hi, teen with IQ 150+ and autism here.
I can't objectively say how high-IQ children behave, because it varies a lot. However, there's a point where we all share the following characteristics:
We have interests/hobbies not shared by most of the population. We tend to have interests and hobbies that most other people our age would find either too difficult or too boring. This often leads to us hanging out alone, as we can only explore these interests with at most a few other people.
We tend to be very perceptive. We can have many ingenious solutions to problems that have long been dealt with in a more traditional way, and we can also have lots of "aha!" moments.
*We have a different perspective. "The more you know, the more you know you don't know." When you're at the top, it's easier to gauge how much you know with how much you don't, helping you set goals with some degree of level-headed-ness.
We can ignore useless information. We're able to assess different situations more logically and can make good executive decisions on what things are relevant or not.
We are observant, and can understand the causes and effects of various actions.
We can keep our thoughts separate from our emotions. This comes in a lot of different forms, such as avoiding debates that reward wins rather than knowledge, being able to accept constructive criticism without breaking down or taking it personally, and making decisions based on fact/truth rather than feelings.
We challenge dogma. If it's not true/correct, it's not true/correct, and we will attack bullshit, no matter what authority tells us to be quiet (whether the government, public, etc.)
We are intuitive. Our minds process information so quickly that we can predict what happens next on a hunch.
We don't like nonsense. (This definitely doesn't apply to all of us, let alone all humans.) Often, we find gossip/trashtalk/cussing contests/etc. to be totally trivial and instead pursue conversations about our interests, which are more substantial to us (meaning that we can come off as anti-social or overly shrewd).
We go beyond applying concepts to problems and can make connections between different concepts or derive the origin of these applications. For example, you can derive the Quadratic Formula and manipulate any of the intermediate forms, rather than just use it to solve for roots of polynomials.
We hate disorder. (Again, this definitely doesn't apply to all of us.) Generally, we hate when people shove others or pickpocket them when they've done nothing. We also hate when other people leave a mess behind them and ultimately leave it to us.
We are very confident, or at least know where to go.
Our humor is somewhat far-removed and requires somewhat obscure knowledge or analysis to understand.
and a lot more.