Does neglecting the Abrahamic prayer in the first tashahhud invalidate the prayer? And what is the ruling on someone who says: "O Allah, bless Muhammad and the family of Muhammad" only in the first or last tashahhud?

The Ibraheemiyyah prayer (Salat al-Ibraheemiyyah) is neither obligatory nor recommended in the first Tashahhud. Omitting it does not invalidate the prayer. Limiting it to "Allahumma salli 'ala Muhammad, wa 'ala aali Muhammad (O Allah, send prayers upon Muhammad, and upon the family of Muhammad)" is sufficient. The pillar of praying upon the Prophet (peace be upon him) is "Allahumma salli 'ala Muhammad (O Allah, send prayers upon Muhammad)."