Is it permissible to sell a gift that the deceased gifted before his death and donate its price on his behalf, and is it permissible to give this money to a mosque under construction?

If you took possession of the gift and acquired it before the death of the giver, there is no blame upon you for disposing of it. You may give it, or its value, as charity on your own behalf or on his behalf. Charity to the mosque is better because it is an ongoing charity (sadaqah jariyah). However, if you did not take possession of the gift before his death, it becomes void upon his death and becomes an inheritance to be divided among his heirs, based on the statement of Abu Bakr to 'Aisha, may Allah be pleased with them both, concerning his gift to her of a date palm orchard: "Had you plucked and taken possession of it, it would have been yours. But today, it is the property of the heirs."