What is the ruling on praying for the Household of Muhammad (peace be upon him) in the prayer, and is it obligatory to pray for them and invoke blessings upon them in the last Tashahhud?

It is not a condition for the validity of praying upon the Prophet (peace be upon him) in the tashahhud to recite the complete Abrahamic prayer. The minimum that suffices is: "O Allah, send Your peace and blessings upon Muhammad."

As for intentionally omitting "the family" (al-aal) from the Abrahamic prayer, it is not permissible, because it was reported this way from the Prophet (peace be upon him).

And according to the view that the prayer upon the family is obligatory, following the Prophet (peace be upon him) (which is a view in the Shafi'i school and a narration in the Hanbali school), then intentionally omitting them invalidates the prayer.

However, according to the view of the majority that praying upon them is not obligatory, the prayer is not invalidated by omitting them.