Learn Finnish: Aiotteko yöpyä hotellissa vai teltassa? - Are you planning to stay overnight in a hotel or a tent?

In Finnish, vai is used exclusively in questions when presenting mutually exclusive options where the listener is expected to choose one over the other. Tai is used in statements, or in questions where the options are not mutually exclusive.

Why are hotellissa and teltassa in the inessive (missä) case? Don't verbs about staying somewhere usually take the mihin case?

You are likely remembering the verb jäädä (to stay/remain), which requires the mihin (illative) case, such as jäädä hotelliin. However, yöpyä (to spend the night) is treated as a static action. Since the entire action happens in one place, it takes the static missä (inessive) case.

Is the verb yöpyä related to the word for night?

Yes, it is derived directly from the noun yö (night). Adding the verbal suffix -pyä turns the noun into a verb meaning "to spend the night" or "to stay overnight."

Why is yöpyä in its dictionary form here instead of being conjugated?

It is in the first infinitive (the dictionary form) because it is governed by the main verb aikoa (to intend/plan). Verbs expressing intent, like aikoa and haluta, always require the action verb that follows to be in the first infinitive.