Is Tsirelson's Bound the only constraint on these quantum correlations?

No. The necessary and sufficient condition is well-known; see for instance Tsirelson 1993, or L. Landau.

No, the converse is not true. Even if a correlation satisfies the Tsirelson's bound, even then it might not be reproducible by measurements on quantum states. One of the most prominent examples is the Hardy correlation. Imposing the Tsirelson's bound, it seems that the maximum value of q is 0.207. However, it turns out that with quantum states, we cannot obtain a q value greater than 0.09.