Learn Turkish: O doktor mu?

The question particle follows 4-way vowel harmony. Look at the last vowel in the word before it, which is the o in doktor. According to the 4-way rules, if a word's last vowel is o or u, the question particle becomes mu.

Does o mean 'he' or 'she'?

It means both! Turkish does not have gendered pronouns. The word o translates to 'he', 'she', or 'it' depending on the context. So this sentence could also perfectly mean 'Is she a doctor?'.

Where is the verb 'is' in this sentence?

There isn't one! In simple Turkish sentences describing what a third person is, the 'to be' verb is usually invisible. The statement O doktor simply means 'He is a doctor'. Adding mu at the end turns it into a question.

Why is mu written as a separate word instead of being attached to doktor?

The yes/no question particle is unique in Turkish. It follows vowel harmony rules exactly like an attached suffix (which is why it matches the vowel in doktor), but it is always written with a space before it.