Why do we invoke blessings upon the Prophet Muhammad (peace be upon him) alone in some instances, and upon "Muhammad and the Household of Muhammad" in other instances? And does the prayer become invalid if we do not mention the Household of Muhammad?

Praying upon the Prophet, may Allah bless him and grant him peace, is a great act of worship for which the doer is rewarded, and it is emphasized when he is mentioned. Outside of prayer, it is sunnah to say: "Allahumma salli ala Muhammadin wa ala ali Muhammad" (O Allah, send prayers upon Muhammad and upon the family of Muhammad). It is not obligatory to mention the Companions, but it is permissible. As for inside the prayer, it is a pillar in the last tashahhud according to the Shafi'is and Hanbalis, and the prayer becomes invalid by omitting it. It is fulfilled by saying: "Allahumma salli ala Muhammad" (O Allah, send prayers upon Muhammad), and the prayer does not become invalid by omitting prayers upon the family.