The do/does form is used when there is no other question word. It is not needed here because "who" is a question word.
All of the names of meats (i.e. beef, mutton, pork, lamb, chicken, venison, etc.) are uncountable nouns. Chicken is unique in that it is both the name of the meat and the animal. If you use the article "a" in English though, it would mean that you ate the entire chicken.
The tips say that question words should be placed at the end of the sentence or right before the verb if there is one. So shouldn't it be: Tavuk kim yer?