Learn Turkish: Bu metni Türkçe okur musun? - Would you read this text in Turkish?

There are two things happening here. First, metin (text) gets the accusative suffix -i because it is the specific direct object of the verb (pointed out by bu). Second, metin is one of those Turkish words that drops its second vowel when a suffix starting with a vowel is added. So, instead of "metini", it becomes metni.

Why use the aorist tense okur musun instead of the present continuous okuyor musun?

While okuyor musun? means "Are you reading?" (an action happening right now), the aorist tense is frequently used to make polite requests. Using okur musun? translates closer to "Would you read?" or "Could you please read?".

Why is there no suffix on Türkçe to say "in Turkish"?

In Turkish, language names like Türkçe or İngilizce already function as adverbs of manner. You do not need to add a locative suffix like -de to say "in" a language. You just place the language word directly before the verb.

Since this lesson is about diye, how would I report someone asking this exact question?

To say "He asked if you would read this text in Turkish," you quote the question exactly as it is and add diye sordu (asked saying). The full reported sentence would be: "Bu metni Türkçe okur musun?" diye sordu.