This does not answer your question. But it was a bit too long to put as a comment.
Firstly, it seems that the following old question is of some relevance.
Families of curves for which the Belyi degree can be easily bounded
In fact, dessins $X\to \mathbf{P}^1$ are also called Belyi maps/morphisms/functions on $X$. I wanted to know of curves for which one has explicit bounds on the Belyi degree, i.e., the minimal degree of a dessin $X\to \mathbf{P}^1$. Here are the examples
- Fermat curves
- Modular curves (congruence or non-congruence)
- Hurwitz spaces (see JSE's answer to the above question)
- Galois Belyi curves = Wolfart-curves = Galois three-point covers
- Elkies' curves (see his answer to the above question).
Let me elaborate on 2. If $\Gamma\subset \mathrm{SL}_2(\mathbf{Z})$ is a finite index subgroup, you can consider the quotient $Y_\Gamma = \Gamma\backslash \mathbf{H}$, where $\mathbf{H}$ is the complex upper half-plane and $\mathrm{SL}_2(\mathbf{Z})$ acts on $\mathbf{H}$ by Mobius transformations. The curve $Y_\Gamma$ naturally inherits the structure of a connected Riemann surface from $\mathbf{H}$. We compactify $Y_\Gamma$ by adding "cusps". The compactification of $Y_\Gamma$ is usually denoted by $X_\Gamma$. Note that there is a natural map $Y_\Gamma \to Y_{\mathrm{SL}_2(\mathbf{Z})} = Y(1)$ induced by the inclusion $\Gamma\subset \mathrm{SL}_2(\mathbf{Z})$. This morphism extends to the compactifications $X_\Gamma \to X(1)$ and induces a dessin $X_\Gamma \to \mathbf{P}^1(\mathbf{C})$ after you compose with the isomorphism given by the $j$-invariant $j:X(1)\to\mathbf{P}^1(\mathbf{C}$. (The branch points are the elliptic points $0$, $1728$ and the cusp $\infty$ of $X(1)$.)
Let me adress your third question. The above is about your second question. I don't have much to say about your first question, unfortunately. What do you mean by a dessin which "captures" all elliptic curves over $\mathbf{Q}$?
Firstly, assume that $X\to \mathbf{P}^1$ is a dessin of prime degree. It's clear that this morphism will not factor.
I get the feeling (but I might be wrong) that you are interested in modular parametrizations of elliptic curves in the following sense. You want to know whether the above explicit dessins on $X_0(n)$ can be shown to factor through some elliptic curve. If this is the case, the answer is likely to be no for $n$ big.
Now, you can bound the number of dessins on a curve $X$ of given degree $d$ by the number of dessins of degree $d$, i.e., the number of topological covers of $\mathbf{P}^1-\{0,1,\infty\}$.
But your $H_{X,Y}$ will be zero or infinite.
In fact, if it not zero then there exists a dessin $X\to \mathbf{P}^1$ which factors through a dessin $Y\to \mathbf{P}^1$. But Belyi proved that for any finite set $B\subset \mathbf{P}^1(\overline{\mathbf{Q}})$ there exists a dessin $R:\mathbf{P}^1_{\mathbf{Q}}\to\mathbf{P}^1_{\mathbf{Q}}$ (defined over $\mathbf{Q}$ even!) such that $R$ sends $B$ to the set $\{0,1,\infty\}$. So from a given factorization $X\to Y\to \mathbf{P}^1$ you can construct an infinite number of really different dessins (and associated factorizations).
The former paragraph is just applying the fact that given a dessin $f:X\to \mathbf{P}^1$ you can construct an infinite number of dessins $g :X\to \mathbf{P}^1$ by composing $f$ with an arbitrary dessin on $\mathbf{P}^1$. (Belyi actually gave an algorithm to compute the above dessin $R$ on $\mathbf{P}^1$ associated to $B$.)
So to make sense of your last "crazy" question, you might want to fix a dessin $X\to \mathbf{P}^1$ on $X$ and try to look at possible factorizations, where $Y$. Thus, let $H_{\pi}$ be the number of pairs $(Y,f)$ (up to isomorphism), where $f:Y\to \mathbf{P}^1$ is a dessin and there exists a factorization $g:X\to Y$ such that $\pi = fg$.
I don't think it is possible to give a precise formula for $H_\pi$ easily, but it is certainly possible to bound this number in terms of the degree of your dessin.