Which region of your brainstem plays a role in arousing you to a state of alertness when, for example, you accidentally stumble over another person's misplaced pair of shoes?

The region of the brainstem that plays a role in arousing you to a state of alertness would be the reticular formation. It is responsible in regulating the sleep cycles and in detecting sensory conditions. It consists of more than 100 neural network that are for various functions.