Have there been any narrations regarding pigeons, and did the Messenger, peace and blessings be upon him, eat in front of a fasting woman, and is the narration in Sahih Ibn Khuzaymah?

There are two meanings of ḥammām in the question:

Al-ḥamām (the bird) – A hadith states that one who excessively follows pigeons is like a devil following a she-devil, particularly if it leads to an obsession with flying pigeons, excessive engagement with them, or climbing rooftops overlooking neighbors' homes.

Al-ḥammām (the bathhouse) – There are hadiths indicating: The entire earth is a place of prayer except for bathhouses and graveyards. A believer does not allow his spouse to enter bathhouses. A believer does not enter a bathhouse without wearing a covering. One should not sit at a table where alcohol is being served.

As for the hadith about the Prophet ﷺ eating in front of a fasting woman, it clarifies that angels send blessings upon a fasting person when others eat in their presence until they finish.