Does Drupal 8 entities have any events I can listen to? I know we can use hook_entity_update() hook to do something when an entity is updated. But is there an event to listen for node updates/deletes, or other entities such for that matter?
You can't listen to an event. You can either use a hook or implement a method like postSave() in your custom entity.

Here is the order of hooks and other events that happen during an entity save:


Entity::preSave() is called on entity objects, and FieldItemListInterface::preSave() on field objects.
hook_ENTITY_TYPE_presave()
hook_entity_presave()
Entity is saved to storage.
Entity::postSave() is called on the entity object.
hook_ENTITY_TYPE_insert() (new) or hook_ENTITY_TYPE_update() (update)
hook_entity_insert() (new) or hook_entity_update() (update)