Learn Turkish: Sözleşmeyi oyuncuya imzalatayım mı? - Shall I have the actor sign the contract?

This is the core rule of Turkish causatives for transitive verbs! The base verb imzalamak (to sign) already takes a direct object: sözleşmeyi (the contract), which gets the accusative case. Because the direct object "slot" is already taken, the person you are making do the action (the causee) gets bumped to the dative case (oyuncuya). You are literally saying something like "Shall I cause signing the contract to the actor?"

How do we form the causative verb imzalatmak here?

It starts with the noun imza (signature). We add -la to make the base verb imzalamak (to sign). Because the verb stem imzala- ends in a vowel, it takes the -t causative suffix to become imzalatmak (to have/make someone sign).

What tense or mood makes imzalatayım mı translate to "Shall I have...?"

It uses the optative mood, which is used for wishes, suggestions, or volunteering. The suffix -ayım means "let me." When you combine this first-person optative form with the question particle mı, it asks for instructions or permission, which perfectly matches the English "Shall I...?" or "Should I...?"

Does it matter that oyuncuya is placed right before the verb instead of sözleşmeyi?