I did not use "dați" since that means "give" rather than "send" ("trimiteți"). Correct?

You can use dați or trimiteți for sending a message.

It does not even give trimiteti (which is suggested when you click on send) as an answer!!

I suppose the correct answer could be "Dati multe bufnițe" if we're talking Hogwarts.

”A da mesaje” is really colloquial Romanian. As the course is about standard Romanian, it should be ”Voi expediați/trimiteți multe mesaje.”

It looks like the whole course was made perfunctory. It is a shame.