Ayasofya seems more likely to be a Turkish transliteration of Hagia Sofia, given it is pronounced exactly the same.
transliteration of Greek "Αγία", to be more specific.
Probably not since a lot of churches start with the name Aya in Turkish because they are named after saints. (Like this one: https://tr.wikipedia.org/wiki/Aya_%C4%B0rini)
Aya Irene is from the Era preceding
Ottomans. That word means "Peace".
Another tidbit: Hagia Sophia can be translated to english as Saint Sofia. It is pronounced in Turkish as Aya Sofia. Aya is from an arabic word that means "sign" like sign of god (similar to "aya"-tullah in Iran). The similarity between "Hagia" and "Aya" is so strong as though it follows the concept of changing the "j" sound to "h" or "y" sound. When applying that same rule to "Ayatollah", you could basically translate the word to "Saint of God". Every time I think of this, it gives me the jitters.