Learn Turkish: Gelin çok ağlamış mı? - Did the bride reportedly cry a lot?

ağladı mı uses the regular definite past tense, simply meaning "Did she cry?". You use ağlamış mı (the reported past tense) when you are asking about something you didn't see yourself. It translates closer to "Did she reportedly cry?" or "I heard she cried, is that true?"

Why does the question particle take the form mı here?

The question particle always follows 4-way vowel harmony based on the final vowel of the word before it. The last vowel in ağlamış is ı, so the particle must be mı. If the verb had been gülmüş (reportedly laughed), the question would be gülmüş mü?.

In what real-life situation would a Turkish speaker ask this specific question?

It is a tradition for a Turkish gelin (bride) to cry when she leaves her family home. Someone who missed the wedding or the send-off might ask this to a guest who was there. Using the -mış tense perfectly captures the feeling of "I wasn't there to see it, so tell me what happened."

Could I put çok at the beginning of the sentence: Çok gelin ağlamış mı?

No, because that changes the meaning completely! In Turkish, adverbs usually go right before the word they modify. Here, çok (a lot) describes how much she cried, so it goes right before ağlamış. If you said Çok gelin, it would mean "Many brides."