Final answer:
A jet injector is used to administer topical anesthesia, and the oral route is the most appropriate for home administration of an antimicrobial to treat a systemic infection.
Step-by-step explanation:
The piece of equipment used to administer topical anesthesia is a jet injector. This device delivers the anesthesia medication through the skin without the use of needles, making it ideal for superficial clinical procedures. On the other hand, when it comes to the routes of administration for treating a systemic infection at home, the oral route is the most appropriate and convenient, as it allows the patient to self-administer the medication.