1) According to the Torah, one of the obligations of a husband to his wife is having marital relations. The discussion in the halachic literature about the minimum frequency of relations includes the talmid chacham, a paradigmatic man who does not work in physical labor and lives at home. His minimum is once a week. Once a weekly requirement is designated, Shabbat becomes a natural preferred choice because of its surpassing holiness. Having relations on Shabbat is a preference, not a requirement.

2) No, a shower is not normally permitted on Shabbat. As in the case of a woman cleaning herself on Shabbat for a hefsek taharah, you may wash the inside of your thighs and genital area with cool water or with water warmed prior to Shabbat and pat yourself dry with a towel, taking care not to wring it. You may also wipe yourself off with a dry cloth.