Is the pilgrim's delay in spending the night in Mina due to exhaustion and lack of transportation considered a valid excuse that removes the obligation of a compensatory sacrifice (dam) from him and his wife, and does spending most of the night in Mina lift the obligation of a compensatory sacrifice (dam)?

What is obligatory for the pilgrim regarding staying overnight in Mina is to spend most of the night there, not to enter it before sunset. If this is fulfilled, then no sacrifice (dam) is required. As for delaying departure for Friday prayer, there is no harm in it for those who are not residents of Mecca, because Friday prayer is not obligatory for a traveler.