To what extent does the overlap of uttering the intention to fast with the Fajr adhan affect the validity of the fast? Is uttering the intention obligatory? Is using the word "tomorrow" in the intention incorrect? Does doubt about the validity of a specific day's fast necessitate its قضائها (making it up)? And what is the authenticity of the presented arguments concerning these issues?

Uttering the intention is not legislated; rather, it is an innovation (bid'ah) according to many scholars. It has not been reported from the Prophet (peace be upon him), his Companions, or the Tabi'un (followers) that they verbally expressed the intention for prayer, purification, fasting, or anything else. The intention is in the heart. Therefore, your fast is valid even if you did not utter the intention, and what matters is what is in your heart, not what you said with your tongue. Doubt about the invalidity of the fast does not necessitate its invalidity.