Let $a$ be a closed point in $\mathbf{P}^1\_{\overline{\mathbf{Q}}}$.


Let $Y \cong \mathbf{P}^1\_{\overline{\mathbf{Q}}} $ and let $\pi:Y\to \mathbf{P}^1\_{\overline{\mathbf{Q}}}$ be a finite morphism which is unramified above $a$. Let $b$ be a point in $\pi^{-1}(a)$. 


Can one effectively bound the height of $b$ in terms of the degree of $\pi$, some data depending on the branch points of $\pi$ and the height of $a$?


Ineffectively this should be possible. 


Maybe it's more natural to ask if one can bound the height of $\pi^{-1}(a)$ in terms of the degree of $\pi$, some data depending on the branch points of $\pi$ and the height of $a$.



Comparing heights of rational points on curves through covers
Let me reformulate the problem in less fancy terms. Let $\pi(x)\in\overline{\mathbb{Q}}$ be a rational function of degree $d\ge1$. As ACL noted, there is a standard estimate
$$
 dh(y) - c\_1(\pi) \le h(\pi(y)) \le dh(y) + c\_2(\pi), \quad (\*)
$$
where it is relatively easy to give explicit formulas for $c\_1$ and $c\_2$ in terms of $d$ and the coefficients of the polynomials defining $\pi$. However, what you seem to be asking is whether we can give formulas for $c\_1$ and $c\_2$ that depend only on $d$ and the branch points of $\pi$. 


I suspect that the answer is no, for the following reason. For simplicity, lets assume that $\pi(\infty)\ne0$, so I can normalize $\pi$ so that its numerator is monic of degree $d$. A rational function of degree $d$ has $2d-2$ branch points, but with this normalization, it has $2d+1$ free coefficients. So fixing the branch points should give a 3 dimensional family of maps with those branch points. By varying $\pi$ within such a family, the values of $c\_1(\pi)$ and $c\_2(\pi)$ that come from the usual proof will not be bounded. This makes me suspect that by varying $\pi$ within the family, the inequality $(\*)$ will fail if we require $c\_1$ and $c\_2$ to be fixed. 


This isn't a proof, of course, but I hope it suggests where one might look for a negative answer to your second question.


