If you keep the commas, you can definitely say "Give me, please, the box". It would function like an independent clause because the sentence still makes sense if you were to remove ",please," entirely.
The sentence might take on a more authoritative emphasis, though. Sort of like, "give me the box, or else!" It's a little weird but I'm pretty certain it is grammatically allowed, provided the commas are there.
Oh my God! Sometimes this strange male robot voices it so funny!
"Дайте, пожалуйста, Коробку́." Ha-ha, Mr. Korobók! (the correct accent: "Дайте, пожалуйста, коро́бку")
"Please give/allow/punch to (Mr.) Box ..."
In Russian, it requires about this continuation:
Please allow (Mr.) Box to make a toast!
Please, punch (Mr.) Box in the eye! :)
Is it possible to switch to a female voice? It talk a little less nonsense.
"box, case; gearbox"
In english when you ask someone to give, you always say "give me" or "give her/him/them", in russian you dont have to include - me, which is мне, only when you want to highlight to whom you want the box to be given. Дайте тебя is a bit meaningless, unless you want to say something like saying someone "give yourself", but in that case you would say дайте себя
"Could you please give me that box" was my translation and it was "wrong". What YOU accept as correct is, your expectations, is over the top. Both English and Polish native speaker here btw. You are wasting people's time by making them retest over and over because they have put words in a different order or forgot an "article". It's ridiculous. I also used to be fluent in Russian as a child, so it's coming back very quickly to me.