I don’t recall who said a child owning a gun is illegal but it’s not here in Illinois. For long guns at least. There is no age requirement at the federal level and at the state level, if you are under 21 you only need a parent’s written consent for a FOID. I can and plan to get one for my 5 year old, not because I intend to turn him lose with one but he should be allowed ownership in the event I or his grandfather want to gift one or will one to him. The responsibility still rests on my shoulders.
For the OP’s questions at hand. I agree, half of your questions aren’t serious. Kids/self defense? Really? Popularity should not be a factor and if you know so little, discussion of guns for competition isn’t even on the table. I am going to assume here that we are talking about a real gun and not an air rifle. If it’s a new shooter, the answer should always be a .22 and should never be left alone with it. I’d recommend something that requires manual intervention to chamber the next round, non-semi auto, like a bolt action, lever action or single shot. They have to learn from the ground up safe handling, operation, aiming and proving they can handle the responsibility of being in control of a gun. The person teaching them needs to already know these things. That’s not you. You need to get some experience under your belt before going forward with putting one in a kids hand.
Never put a semi auto anything in the hands of someone who doesn’t know what they are doing or what to expect. Each time one of my boys graduates to a bigger gun I make sure they can handle the load and experience the shock associated with the round before even thinking about handing them something that will auto chamber the next round. This means if it’s a semi auto, I load 1 round at a time until We are both comfortable with how he handles it. Then we move to 2 rounds, etc. I’ve seen experienced adult shooters double tap the trigger because they were startled or didn’t have a good enough grip when it was fired and even drop the damn thing, loaded and ready to fire. Seriously, do a YouTube search on gun fails, it’s quite common. Never do that to a kid. It’s the equivalent of giving someone who’s never driven before, the keys to your corvette and expecting that nothing bad will happen.
Finally, you need to be intimately familiar with their behavior. What’s their attention span, do they follow instructions well and responsible or are they careless and/or defiant? I’d take a responsible 7 year old who takes instruction well to shoot long before I’d take a 15 year old who doesn’t listen. Handing someone a gun is a huge responsibility for both of you. If they hurt themselves or someone else, it’s on you. You’re the adult.