Learn Turkish: Kadının çocukları uyuyor mu? - Are the woman's children sleeping?

It uses a definite noun compound. The owner gets a genitive suffix (kadın + -ın = kadının, meaning the woman's). The owned item gets a possessive suffix (çocuklar + -ı = çocukları, meaning her children). Together they form kadının çocukları.

Why does çocukları end with an -ı instead of -i?

It follows 4-way vowel harmony. The possessive suffix looks at the last vowel of the word it attaches to. Because the last vowel in çocuklar is an a, the suffix becomes -ı.

Since the children are plural, shouldn't the verb be uyuyorlar instead of uyuyor?

It can be, but it is optional! In Turkish, when a plural subject is already clearly stated in the sentence (like çocukları), it is very common and natural to leave the verb in the singular form (uyuyor).

Why is the question word mu used here instead of mı?

The question particle also follows 4-way vowel harmony, matching the last vowel of the word right before it. The last vowel in uyuyor is o, which means the particle must be mu.