When pressing Q/Ctrl+Q without items. Both single and stack. (tested in sp).
May be intended.

I think its intended cos it still sends a packet to the server, then realises there is nothing to drop. I may be mistaken.

Didn't know about it, but the docs don't say any different as it is:

Triggered when the game receives the request from a player to drop one item or full stack from its inventory

I thought to update the docs, but the docs are right.

That does make sense, yeah, but since there is not many real feedback from it (it just tells who tried to), I thought that at least having it only trigger if the player is really dropping something would make it more useful.
Not because I really use it, I just found it when testing event_test for Scarpet rules and thought it would be nice to report it.
Feel free to close it if you think it should stay.

yeah, this is very much so intended - gives much more abilities for limited inputs that server gets from clients.