Yes. You do not go back to being virginal even if all of the antibodies from previous infection have disappeared or weakened, because the T-cells have downloaded and contain the "software" for creating new antibodies at a much faster rate and in a target specific way. Whereas the first time around the body has to cast around with trial and error.
Similarly, some people will argue that a flu vaccine or a previous flu infection is worthless as tits on a nun because people still get it year after year with or without the vaccine or regardless of whether they have it before.
Yes and no. Yes, in that the immunity is often not sufficient to deal a knock out blow to the virus such that it is immediately neutralized and so therefore you have to deal with a mild case of it. No, in the sense that it definitely provides massive protection from mortality among everyone who has at least some remnant of an immune system left. The flu and the common cold are absolutely and totally lethal to people who have had no previous exposure. You can wipe out half the tribes in the Amazon with a virus where they have had no exposure. So on a very significant level, even partial immunity is enormously beneficial.