Are sums of money paid or gifts offered to employees to facilitate transactions and expedite the disbursement of entitlements considered bribery? What is the Sharia ruling on what a father does in this regard, and how should a son act?

Bribery is one of the major sins. It is what is given to a judge to rule unjustly, or to gain a position of authority, or to oppress a person, or to repel a right, or to obtain something false. If what your father pays is to attain his right or what he is more entitled to, then there is no objection to that, and the sin is on the one who takes it. However, if the purpose is to seize the right of others or to obtain something in which he and others are equal, then it is not permissible.