1. (a) That's wrong. A person is entitled to know with whom they are entering into legal relations. If an agent makes it known that they are acting as a representative of a principal (and if the principal is either identified or identifiable as a particular individual) the principal is said to be 'disclosed' to the third party. When a person enters into a contract on behalf of a disclosed principal, the likely inference is that the agent does not intend to become a party to the contract - they are acting in the place of the principal. Accordingly, the contract that is created exists between, and is enforceable by, the principal and the third party only.
In the present case, A has made it clear when signing the contract that he is doing so as an agent, on behalf of his company, which is the principal.
The inference that an agent for a disclosed principal does not intend to be a party to the contract can be rebutted if, in the circumstances, it can be shown that the agent intended to be personally liable on the contract (in addition to the principal). No such facts exist in the present case.
If a disclosed principal does not actually exist, a person who ostensibly contracts only as the agent of that principal will not be personally liable on the contract. This only applies where the agent and the third party did not actually know, when contracting, that the principal does not exist.