$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$ We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ which is holomorphic on the upper half plane $\mathcal{H}$. Let $f$ be a modular function for $\Gamma$, and suppose $f$ is not a modular function for any larger subgroup of $\text{SL}_2(\mathbb{Z})$.
We may assume that $f$ is integral over $\Qbar[j]$, in which case by Does a modular function primitive for $\Gamma$ generate the function field of $\mathcal{H}/\Gamma$? , its monic minimal polynomial has degree equal to $n := [\text{PSL}_2(\mathbb{Z}):\pm\Gamma]$.
The extension $\Qbar[j][f]$ over $\Qbar[j]$ is in general not etale. This is bad for me if the branch locus contains points where $j$ is not an algebraic integer, so I'm interested in using $f$ to construct other modular functions $f'$ such that $\Qbar[j][f']$ is branched only above algebraic integers on the $j$-line. I will say that a modular function $f$ is only branched over algebraic integers if $\Qbar[j][f]$ is only branched above algebraic integers on the $j$-line.
Now fix an $f$ a modular function for $\Gamma$ as above.
Question 1: Does there exist a $\gamma\in\text{GL}_2^+(\mathbb{Q})$ such that $f(\gamma z)$ (as a modular function for $\Gamma\cap\Gamma(N)$ for some $N$) is only branched above algebraic integers?
Question 2: Do there exist congruence modular functions $g,h$ such that $gf+h$ is only branched above algebraic integers?
Question 3: Does there exist a congruence subgroup $\Gamma_0$ such that if $\Qbar[\Gamma_0]$ denotes the affine ring of $\mathcal{H}/\Gamma_0$, then $\Qbar[\Gamma_0][f]$ is unramified over $\Qbar[\Gamma_0]$ away from the preimages of $j = 0,1728$?
I sort of have no intuition for this - In part this is just a commutative algebra question - what options are there for "twisting" $f$ into some $f'$ so that the discriminant of the minimal polynomial for $f'$, as a polynomial in $j$, has only algebraic integer roots?
