Yes.

Yes, but sometimes I skip it. It depends on the reason why you are applying a second coat. Is it because the new paint color didn't cover the old color? If the old color is still showing through along the cut-in areas, then a second coat is needed. If you are painting over a flat paint with a shiny paint, the often the sheen will not look even unless you apply a second coat everywhere. You will see a difference in sheen between the cut in areas and the rolled areas. Sometimes this is hard to see though depending on lighting. Two coats might even be required if you are painting with one brands eggshell over another brands eggshell. This is why we say two coats is best though in general...there are too many factors that affect that decision.