Am I understanding right that this (“Дайте, пожалуйста…”) is a normal way to order your drink in a café? If so, could it accept some idiomatic translation of that in English (e.g. “Could I have…” or “I’d like…”), since the literal translation (“Give me … please”) is rather unidiomatic?
No, it is fine. In Russian, imperative + пожалуйста is considered polite enough for most situations—at least, when the listener will happily cooperate.
I daresay, structures like "I would like" or "Could you please" are even a bit over the top in Russian. You would definitely use them in formal correspondence ("Could you clarify that last point, please?") but rarely in any real conversation where you expect cooperation. For example, "Could you please pass me some bread?" literally translated would surely be too much for such a simple request (in Russian).
I think I have read a research on these structures. Even some Slavic languages are different.
In Russian, though, please+imperative is acceptable and produced the desired result. At the same time, flowery "would you do smth" and stuff sometimes caused listeners to misinterpret the request, especially if it did require some minimal effort (apparently, sometimes they assumed they were supposed to maybe do it in the future).