(a) No. The creation of a legally binding contract can be delayed if that is what the parties intend. For example, when it is agreed by the parties that their agreement is subject to a formal contract being drawn up and signed, the surrounding facts may indicate that they intend to avoid being legally bound by their agreement until those formalities are completed.
In other circumstances the facts may show that only performance of the agreement is intended to be delayed until some future event occurs. In such cases a legally binding agreement is created even before the event in question takes place. Great care must thus be taken in each case to establish exactly what was intended.
In the example, it seems likely on the facts that A did not intend to be bound by the agreement until a written contract was signed. He did not intend only to delay the performance of the agreement.