Is it permissible to perform communal Takbir after the Eid prayer in the mosque, while appending "peace and blessings be upon the Prophet and his family, companions, supporters, wives, and descendants," as is done in some mosques in Egypt?

Collective takbir (proclaiming God's greatness) on the two Eids is permissible. The wordings for takbir are broad, and the most authentic one reported is "Allahu Akbar, Allahu Akbar, Allahu Akbar kabira" from Salman. It is also said that one can add "La ilaha illa Allah wahdahu la sharika lah..." As for adding blessings upon the Prophet - peace be upon him - after the takbir, some scholars deemed it good, as it does not contradict what has been reported, while others disagreed. The more preferable approach is to omit it and restrict oneself to what has been reported from the Pious Predecessors (Salaf).