(a) That's right. Many people are employed to do particular tasks for their employer, without also being given the authority to act as their employer's agent.
What makes an employee also an agent of their employer is the grant to the employee of an agency power, that is, the power to represent the employer in the conduct of the employer's legal relations.
The question is whether the facts show that Tyler had given Paul an authority to carry out legally binding transactions of his (Tyler's) behalf, and this seems to be so. In particular, Tyler instructs Paul to negotiate joint venture agreements for a country-wide tour of the show.
Joint ventures involve contracts, and Paul is given authority to bring about such contracts between Tyler and the theatre owners. Clearly, Paul is vested with an express agency power.