Define children, are we talking about little toddlers who parents annoyingly spam especially newborns all over their facebook pages? While it irritates the fuck out of me, no I don't think it should be banned they're only children. They're going to end up looking so different when they're older compared to what they are now.
As for teenagers and so on, I don't know about illegal but the teenagers in question should somehow be able to file a request perhaps? I think that does potentially fall within violation of privacy and so on if the parents just don't fucking care and upload anything for anyone to see.
I do find it remarkable that within a society teenagers in particular seem to have fuck all rights despite being individuals perfectly capable of free thought.
Yes and no. That frontal lobe doesn't fully develop until 25 years of age in males.