What is the ruling on saying "Amen" after the Abrahamic prayer outside of formal prayer, and is it better to send blessings upon the Prophet (PBUH), or to say "Amen" upon hearing it, or to combine both?

It is permissible to say "Amin" (Ameen) after invoking blessings upon the Prophet, peace and blessings be upon him, because it is a supplication, and Al-Muhibb Al-Tabari explicitly stated this. It is preferable for a Muslim to invoke blessings upon him when they hear someone else invoking blessings upon him, due to the encouragement to do so and the warning against neglecting it, as the Prophet, peace and blessings be upon him, said: "The miser is he before whom I am mentioned, and he does not invoke blessings upon me." There is no harm in combining saying "Amin" and invoking blessings.