Given a strictly positive integer $A$, let $D(A)$ denote the set of all real quadratic algebraic numbers with a continued fraction having almost all coefficients $\leq A$.

Consider the field $Q_A$ generated by all elements of $D(A)$. One has $Q_1=\mathbb Q[\sqrt{5}]\subset Q_2\subset Q_3,\dots$.

The inclusion $Q_1\subset Q_2$ is strict since $Q_2$ contains for example $\sqrt{2}=[1;2,2,2,\dots]$ and $\sqrt{3}=[1;1,2,1,2,\dots]$.

Are there other strict inclusions? Are there cases of equality? (I ignore for example if $Q_2$ is a proper subfield of the field $\mathbb Q[\sqrt{\mathbb N}]$ generated by all real quadratic number-fields.)

A related question: Given a real quadratic algebraic number $\alpha$ with continued fraction expansion $\alpha=[a_0;a_1,a_2,\dots]$ consider the mean value $\mu(\alpha)=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^na_j$. Since $[a_0;a_1,\dots]$ is ultimately periodic, this mean value is a well-defined rational number $\geq 1$ if $\alpha$ is irrational.

Are there examples of quadratic number-fields $\mathbb Q[\sqrt{N}]$ such that $\inf_{\alpha\in \mathbb Q[\sqrt{N}]\setminus\mathbb Q}\mu(\alpha)>1$?