Depends on the child. I was 3 when my dad decided I understood the safety rules and let me use a BB gun to demonstrate that knowledge. I was 5 when he started letting me shoot single-shot .22, but that was because I had been consistent with the safety rules for almost 2 years with the BB's and could finally reach the trigger on his .22 he'd had as a boy. I was 6 when he bought me my first .410. 7 was my first 20 ga. semi-auto. At 9 replaced the single shot with a model 60 Marlin.
--One of my daughters was barely 4 when she could follow the rules and started with the BB's. Another was almost 6. Another is still a bit untrustworthy on safety at age 11, but we're working on it (very carefully). I now have two daughters who are too small to reach the trigger on almost any youth model bigger than a crickett, but both are competent shooters. I mounted a 10/22 into an axiom adjustable stock (9 1/2" - 13 1/2" LOP) and put a Harris bipod on the front to help them deal with the weight. The smallest one likes to shoot 1" diameter swinging plates until she obliterates the stickers. Then it's "time to get more stickers!" One note on very young shooters. Daddy handles all ammo, loading, reloading etc. to cut down on lead exposure. I have outdoor ranges. So, leap vapor isn't as much of a concern.
9mm and 45acp are equally lousy. There's not a dime's worth of difference in REAL terminal ballistics. Get over it.