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.

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). (MathSciNet)

He proves that for $1 < \lambda < 2$, for almost every $x$, the set $\mathcal{Q}(x,\lambda)$ grows like $n^\frac{1}{2-\lambda}$. More precisely, Erdos proves that a one-sided version of your set $$\mathcal{Q}'(x,\lambda) = \big\{ q \geq 1 \ \big| \ \text{there exists } p, \ (q,p) = 1, \text{ such that } 0 < x-\frac pq < q^{-\lambda} \big\}$$ satisfies $$\lim_{n \to \infty} \frac{\big |\mathcal{Q}'(x,\lambda) \cap [1,n] \big| }{\sum_{q=1}^n q^{1-\lambda}} = \frac{12}{\pi^2}.$$ That the set $\mathcal{Q}(x,\lambda)$ grows like $n^\frac{1}{2-\lambda}$ 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.