Problem 1 is solved completely, in the affirmative, in the following paper of Grosswald:
Emil Grosswald, Some theorems concerning partitions, Trans. Amer. Math. Soc. 89, 1958, 113–128.
Grosswald in fact gives a very accurate estimate for the asymptotics of q_R(n), $q_R(n)$, showing that they grow exponentially fast with n, $n$, and generalizes things to the case that R consists of any finite union of arithmetic progressions as well. (If you look at his rather intricate paper, look at the function that he calls $H(x)$).