1. (a) That's wrong. Generally, an agent can be appointed and given authority orally, without any written formalities. But there are some exceptional cases. One is where an agent is appointed for a period exceeding a year, or to do an act which will not be completed within a year. In such cases, the appointment of the agent must be evidenced in writing. Generally, an agent must also be appointed in writing if they are to buy or sell land, or an interest in land, or a business.
A written document appointing and empowering an agent to represent their principal is called a 'power of attorney'. There is legislation in the various states governing various aspects of powers of attorney, for example, their form and content.