1. (a) That's probably right. A makes it clear to C that he is not buying the cabinet for himself, so it cannot be inferred that he intended to be bound by the contract to buy the cabinet. However, A might be liable to C on a 'collateral' contract (often called a warranty of authority).

This can happen if an agent discloses that they are acting for a principal, and expressly or impliedly guarantees that they have been given the authority needed to enter into the transaction on behalf of the principal. If it turns out, despite giving this guarantee, that in fact they have no such authority, the agent may be liable in contract to the third party for a breach of their warranty of authority. They would then be liable for losses flowing from that breach.

The facts in this case strongly suggest that A has impliedly given a warranty of authority, making him liable to C for the difference between the price he agreed to pay C and the price obtained when the cabinet is resold after B repudiates the contract.