Learn Turkish: Hem bulaşık yıkıyor hem şarkı söylüyor muydu? - Was she both washing dishes and singing a song?

Hem ... hem is a common pair of words that means 'both ... and'. It connects the two actions to show that she was doing them at the exact same time: both washing dishes (hem bulaşık yıkıyor) and singing (hem şarkı söylüyor).

Since there are two actions happening in the past, why is muydu only at the very end?

In Turkish, when you connect two verbs with the exact same tense and question ending, you can drop the ending from the first verb to avoid repeating yourself. Instead of saying yıkıyor muydu ... söylüyor muydu, you just put the verbs in the present continuous tense (yıkıyor, söylüyor) and add the past question particle (muydu) to the last one. It applies to both actions!

How exactly is the question word muydu built?

It breaks down into three parts: mu is the question particle (matching the rounded vowel in söylüyor), y is a buffer letter to keep two vowels from crashing together, and du is the past tense ending ('was'). Together, they change 'is she singing?' (söylüyor mu?) into 'was she singing?' (söylüyor muydu?).

Why are the words for dishes (bulaşık) and a song (şarkı) not plural or using the word bir?