Translation:In which hotel are the tourists spending the night?
Your assumption is correct, no word for "spending" is needed. The translation is as shown. It's somewhat idiomatic, similar in English to saying "in for the night." In case you are interested, Finnish does have another verb "yöpyä" which specifically means "to stay overnight" (minä yövyn, sinä yövyt, hän yöpyy... etc.), but this course does not include it in the curriculum. There is also "viettää", meaning to spend time; I believe that does get covered in this course.
Couldnt this also be translated as "where in the hotel are the turists spending the night?" I.e. which room are they staying in? To my mind, there isn't anything in the finnish sentence to suggest you are referring to multiple different hotels they could be in. I'm probably wrong but would like to understand the difference
A native speaker should weigh in here, as they could better explain the nuances, but my understanding is that "missä hotellissä" is taken strictly to mean "in which hotel" (or more colloquially "what hotel", as in "what hotel are they staying in") because the inessive case is required for conveying that. To express what you are suggesting (a specific place in the hotel), there are probably a few ways to do it. My vote would be "Missä tässä hotellissa turistit ovat yötä?" or "Missä huonessa turistit ovat yötä?" Hopefully, this makes sense.