Learn Turkish: Baban salonda kanepede oturuyor muydu? - Was your father sitting on the couch in the living room?

This is the past continuous question form! It takes the verb root otur (sit) and adds the continuous suffix -uyor. Then, it separates the question particle mu, adds the buffer letter y, and attaches the past tense -du. Together, they form 'was he sitting?'.

Why are there two location suffixes used back-to-back in salonda kanepede?

Turkish often lists locations from the largest area to the smallest. Salonda puts the action in the living room, and kanepede narrows it down to the couch. Even though English says 'on the couch in the living room', Turkish prefers this zooming-in order.

Where is the word for 'your' in this sentence?

It is built right into the word baban. The letter -n added to baba is the possessive suffix for 'your' (singular informal). You could say senin baban, but Turkish usually drops the word senin because the suffix already makes the owner clear.

Why does the question particle take the form muydu here instead of miydi?