Yes, conversationally I almost never here "的" in this type of situation. I think they started out with almost all of these sentences rejecting it if you left it out, but now they seem to have corrected most of them.

的 should be optional if its a Confucian relationship? Like a child and parent...

Yes. This needs to be corrected.

I know famial relationships can be omitted, which applies to things like sons or daughters. I'm not sure if it applies directly to children.

Hmm, I agree that "你孩子在桌子下" should be accepted too...

From what I know, you are quite right because just "下" means directly under, touching the object (e.g. a newspaper under a book), and "底下" means under the object but with some space in between (e.g. under a table/chair/tree).

But maybe it's also regional and can be interchangeable, I have no idea :)

From what I know, you are quite right because just "下" means directly under, touching the object (e.g. a newspaper under a book), and "底下" means under the object but with some space in between (e.g. under a table/chair/tree).

But maybe it's also regional and can be interchangeable, I have no idea :)