An answer to your second question may be found in an old paper of Erdos: Some results on diophantine approximation. Acta Arith. 5 1959 359–369 (1959).

He proves that for $1 < \lambda < 2$, for almost every $x$, the set $\mathcal{Q}(x,\lambda)$ grows like a polynomial of degree $\frac{1}{2-\lambda}$.  This is consistent with the fact that $\mathcal{Q}(x,1)$ is all of $\mathbb{N}$ while the numbers in $\mathcal{Q}(x,2)$ coming from continued fraction convergent denominators grow exponentially.