Coronaviruses are called so because they look a certain way under an electron microscope. They don't necessarily have common proteins. So you point a does not stand. As for point b,due to the fact that antibodies are specialized and there are tons of viruses that cause the common cold, I wouldn't expect them to match or even exist at any given time in the bloodstream since they only last ~6m and after that memory takes the role of rapid production of new antibodies in case of a new infection.