Learn Turkish: Arkadaşın akşam mısır yiyor muydu? - Was your friend eating corn in the evening?

It is made of two separate parts. The first is the verb with the present continuous ending: yiyor (eating). The second is the question particle mu, joined by the buffer letter y to the past tense ending du (was). Together, yiyor muydu means 'was he/she/it eating?'.

The verb for 'to eat' is yemek, so why does it become yiyor instead of yeyiyor?

When you add the continuous suffix -iyor to a verb stem that ends in the vowels a or e, that vowel narrows. For yemek, the stem is ye-. The e changes to an i, making it yiyor.

Why doesn't akşam have an extra ending to mean 'in the evening'?

In Turkish, common time of day words like akşam (evening), sabah (morning), and gece (night) can be used as time adverbs exactly as they are. You do not need to add a locative ending like -da to say 'in the evening'.

Could arkadaşın also mean 'your friends'?