Let $α$ be an irrational number, let $X_1, X_2, \ldots$ be independent, identically distributed, integer-valued random variables, and put $S_k=\sum_{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails $P (|X_1|>t)\sim ct^{-β}$, $0<β<2$, in Part I of this paper we proved that, up to logarithmic factors, the order of magnitude of the discrepancy $D_N (S_k α)$ of the first $N$ terms of the sequence $\{S_k α\}$ is $O(N^{-τ})$, where $τ= \min (1/(βγ), 1/2)$ (with $β=2$ in the case of finite variances) and $γ$ is the strong Diophantine type of $α$. This shows a change of behavior of the discrepancy at $βγ=2$. In this paper we determine the exact order of magnitude of $D_N (S_k α)$ for $βγ<1$, and determine the limit distribution of $N^{-1/2} D_N (S_k α)$. We also prove a functional version of these results describing the asymptotic behavior of a wide class of functionals of the sequence $\{S_k α\}$. Finally, we extend our results to the discrepancy of $\{S_k\}$ for general random walks $S_k$ without arithmetic conditions on $X_1$, assuming only a mild polynomial rate on the weak convergence of $\{S_k\}$ to the uniform distribution.