Does a Muslim receive reward and protection from evils if he recites the adhkar that have a narrow timeframe—such as the adhkar for entering and exiting the mosque, before and after ablution, and for riding in a car—whenever he remembers them, even if their time has passed?

The adhkar (remembrances) that are tied to specific occasions: if a Muslim forgets them and then remembers them, it's either before the occasion has passed, in which case it is recommended for him to say them; or he remembers them after the occasion has passed. These adhkar are missed when their occasion passes, and it is not legislated to make them up.

Ibn Allan said: "The adhkar related to occasions... it is not recommended to make them up when their occasion has passed."

Shaykh Ibn Uthaymeen said: "Voluntary acts of worship (nawafil) are of two types: those that have an occasion, and those that do not. The one that has an occasion is missed when the occasion passes, and it is not made up."

If these adhkar are missed, they can be compensated for by general remembrance (dhikr) and supplication (du'a).