what can an adult do with Family Sharing? I'm told that some children have iPhones, that are part of 'the same iCloud' as a parent. I've seen that their messages appear to be mixed in with the parent's: the child's message to a friend is replied to both by that friend and their parent.

Can anyone explain how this works?  Is this Family Sharing?  Can the adult account view all communication (e.g. WhatsApp, Skype, Discord) on the linked children accounts, or just that on Apple apps?  Can the adult delete photos on the linked accounts?
The scenario you describe is what Family Sharing is designed to avoid. 

Each person should have their own iCloud account, then each person's data is exclusive to them, but apps/books/music/calendars can, at the parents' discretion, be allowed to be used on the children's devices.
That's just the apps, not the data on those apps.