Is setting a specific value for gifts at a family gathering, then distributing them by random draw to winners of competitions, considered gambling, given that all expenses for the gathering are paid equally by the families?

There is no harm in holding the celebration and competitions in the manner described, as long as they are free from Sharia violations. There is no difference between drawing for an unspecified or a known gift if it increases a Muslim's understanding of their religion or provides permissible knowledge, provided that the gifts are donations and that it is not obligatory for all participants to bring gifts. Rather, it is sufficient for the head of the family to bring one gift on behalf of their family members.