Is a car accident predestined and written in the Book, and is it part of the belief in Divine Decree and Destiny, or is it due to negligence and a failure to take precautionary measures, or a trial to test patience, or a sign of God's wrath?

Everything that occurs in the universe, whether good or evil, was preceded by Allah's knowledge and inscribed by the Pen in the Preserved Tablet. There is no contradiction between events being by Allah's divine decree (qada') and preordainment (qadar), and their being a consequence of the servant's shortcomings, for all of that is by Allah's decree. The wisdom behind these events is to expiate sins, or to elevate ranks, or to test the believers, or as a punishment for misdeeds.