Generally, yes, it's illegal to brew if you're not of age. Just like it is to drink it, but we know how that goes.
I'm not aware of any state that allows you to sell homebrew. Since it's a food product, you have to have permits for legality.
Yes, I think. Giving it away is fine, but you're still liable for possession of it, so...
That's fully dependent on where you live, however you're still liable for possession. There's some states where you can BYOB and there's others where you can't. Again, though, by law you can't possess alcohol.
How to stay out of trouble? Don't be stupid. Brew away, son, but don't try to sell it and don't throw ragers that get the cops called. If you brew beer and drink in peace in your home, no one's gonna know. There's several states where it's technically illegal to brew beer, but that doesn't stop homebrewers.
Again, just don't be stupid. Just like any party I've gone to at a frat or apartment, sell a cup for $5 and provide the beer. Just. Don't. Be. Stupid. About. It.