Learn Turkish: Yakışıklı damat gülümsemiş mi? - Did the handsome groom supposedly smile?

Using the regular past tense gülümsedi mi? just asks "Did he smile?" as a simple fact. By using the reported past tense -miş with the question particle mi, you are asking about something you didn't see yourself, like a rumor or a story. It translates closer to "Did he supposedly smile?" or "Is it true that he smiled?"

How is the verb gülümsemiş related to the word for "laugh"?

The root verb for "to laugh" is gülmek. Turkish creates the verb "to smile" by adding a suffix to this root, giving us gülümsemek. In this sentence, we drop the infinitive -mek and add the reported past tense suffix -miş.

Why is the question particle mi and not mü, since the verb root has "ü" in it?

The question particle always follows 4-way vowel harmony based on the very last vowel in the word right before it. Even though gülümsemiş has ü and e earlier on, the last vowel is the i in the -miş suffix. Therefore, the question particle becomes mi.

Does damat only mean "groom"?

damat means "groom" (the man getting married), but it is also the standard Turkish word for "son-in-law". You have to rely on context to know which one is meant, though "handsome groom" is a very common pairing!