You are referring to the doctrine of Respondeat Superior, where a principle may be liable for the actions of his/her agent. The answer is, it depends on the action, the employment arrangement, and...
It depends on the locality. In some places they go on file with the department, in some places (usually small cities) the officer keeps them and probably eventually throws them away.
For some things -yes. He is a part of your business. Now if you sell shoes and he sold a car or got a girl pregnant That was not in your hiring booklett.