Is there any standard notation for specifying dimension of a matrix after the matrix symbol?

Sometimes your notation is used. The problem of the ambiguity of $X_{m \times n}^T$ comes from the wrong usage. There is no such thing as $X_{m \times n}^T$. Instead, there is $X_{m \times n}$ and there is $X^T$. Combining these two will give you either $(X_{m \times n})^T$ or $(X^T)_{m \times n}$, and removing the parentheses implies that $*_{* \times *}$ and $*^T$ commute (like, for example, $X^{-T} = (X^{-1})^T = (X^T)^{-1}$), which is wrong.

However, I see very little, if any, practical usage in this kind of typesetting. More or less standard way is $X \in \mathbb{R}^{m \times n}$ or $X \in M_{m \times n}(\mathbb{R})$, and then you just use $X$. Writing dimensions in the formulas might make sense at the very beginning of learning this stuff, but not for long, and in this case, parentheses also make a lot of sense.

For anything more advanced, let me give you an example: would you, in a similar fashion, write

$$x_\text{even} + y_\text{odd} = z_\text{odd}?$$

Properties of the objects usually unnecessarily clutter your formulas, and are far better to be declared before the first use, instead of all the time. IMO, of course.

We've always used $$\boldsymbol{A}\in\boldsymbol{M}_{m\times n}(\mathbb{R})$$ to say that it's a real-valued matrix of size $m\times n$. So I guess a more general form would be $\boldsymbol{A}\in\boldsymbol{M}_{m\times n}(F)$ for a matrix with entries in some field $F$.

As requested, the subscript $m \times n$ notation is perfectly common, if not standard.

Of course, we can always use parenthesis to resolve any difficulties. It should be clear that $$\left(A_{m\times n}\right)^T = \left(A^T\right)_{n\times m}.$$

Of course, with a simple operation like the transpose, it looks much nicer as just $A^T$, or, if necessary $\left(A_{m\times n}\right)^T$. The dimensionality of the resulting transpose should be obvious.

Nevertheless, it's not bad practice especially for teaching to remove any possible ambiguities when doing several matrix-matrix or matrix-vector multiplications.