(a) That's right. When a principal has not been disclosed by the agent, the third party will believe that they are entering into an agreement with the agent. The law gives effect to this belief and the agent is made personally liable on the contract. Once the existence of the principal becomes known, the third party has a choice whether to enforce the contract against the agent or the principal.
If the third party makes an unequivocal choice to hold the principal liable on such a contract, rather than the agent, then the right to sue the agent is lost. When and whether such an election has been made is a matter to be determined on the facts of the case.
In the present case, if A signed without disclosing the existence of the company, he would be liable on the contract as a principal contracting party, and B would be entitled to sue him rather than the company.