Is it permissible to declare one's intention aloud or to pray on behalf of another (in a congregational prayer), except in two agreed-upon cases?

It is not prescribed to utter the intention aloud or silently. Uttering it aloud is a more severe innovation (bid'ah) because it has not been authentically reported from the Prophet, peace and blessings be upon him, nor from his Companions. An exception to this is uttering the intention for Hajj and Umrah, but this is not truly an utterance of intention; rather, it is a manifestation of the ritual, similar to the takbirat al-ihram for prayer.

As for deputizing someone to pray on one's behalf, it is not permissible, because prayer is a purely physical act of worship, except for the two rak'ahs of Tawaf which are ancillary to Hajj or Umrah. It is permissible to donate the reward of acts of worship to the deceased, and this includes prayer, but this is not a matter of deputization.