Learn Turkish: Bu, sadece inanılmaz bir tesadüf olmalı. - This must be just an incredible coincidence.

You are right that the -malı / -meli suffix is usually taught first as an obligation ("must" or "should"). However, just like in English, it is also used for strong deduction or probability. Here, olmalı means "it must be" in the sense of "I strongly deduce it is," not "it has an obligation to be."

Why is bir placed in the middle of inanılmaz bir tesadüf instead of at the beginning?

In Turkish, when you have an adjective and the indefinite article bir (a/an), bir always goes directly before the noun, after the adjective. So "an incredible coincidence" becomes inanılmaz bir tesadüf (literally "incredible a coincidence").

What is the purpose of the comma after Bu?

The comma clarifies that Bu (This) is the standalone subject of the sentence. Without it, Bu might look like a demonstrative adjective modifying the next word. The comma creates a slight pause, showing exactly where the subject ends.

Is inanılmaz an adjective or a verb here?

It functions as an adjective meaning "incredible" or "unbelievable." However, it is built from the verb inanmak (to believe). The -ılmaz ending is a negative aorist suffix that turns verbs into adjectives, literally meaning "that which cannot be believed."