Learn Turkish: Peri hangi ormanda yaşamış? - In which forest did the fairy live?

In Turkish, fairytales, myths, and legends are almost always told using the -mIş tense. Since a peri (fairy) is a mythical creature and the speaker didn't personally witness the events, the reported past is the most natural choice.

Why does the verb root yaşa- take -mış rather than -miş or -muş?

This follows 4-way vowel harmony. You look at the final vowel in the verb root yaşa-, which is an 'a'. Vowels 'a' and 'ı' take an 'ı' in the suffix, making it -mış.

Why does the 'in' suffix (-da) go on orman instead of the question word hangi?

Hangi (which) acts as an adjective describing the noun orman (forest). In Turkish, location suffixes like the locative -da (in/on/at) attach directly to the noun, leaving the adjective in front of it unchanged.

Could I put hangi ormanda at the very beginning of the sentence?

Yes, Hangi ormanda peri yaşamış? is grammatically correct. However, in Turkish, the specific piece of information you are asking for is usually placed right before the verb for emphasis. Putting it right next to the verb emphasizes 'in which forest?'.