Certainly not. Ogg proved in "Real points on Shimura curves" that there are only finitely many Shimura curves (of the form $X^D_0(N)$) which are hyperelliptic. If there is a prime divisor $q$ of $D$ such that $q\equiv 1\bmod 12$ then for all $N$ coprime to $D$, $X^D_0(N) \to X^D_0(1)$ is a connected finite 'etale cover.

Now take $D = 26$. $X^{26}_0(1)$ is of genus 2 and hence is hyperelliptic. Take $N$ large enough and you're done.

I also guess there are easier examples but this was the first one to come to me.

edit: This is a counterexample for your second question as well because the cover in question is 'etale but not Galois in general (say for $N$ large enough). If the whole cover $Y \to X \to \mathbb{P}^1$ were Galois, then we'd have $Y \to X$ be a Galois cover as well.

Certainly not. Ogg proved in "Real points on Shimura curves" that there are only finitely many Shimura curves (of the form $X^D_0(N)$) which are hyperelliptic. If there is a prime divisor $q$ of $D$ such that $q\equiv 1\bmod 12$ then for all $N$ coprime to $D$, $X^D_0(N) \to X^D_0(1)$ is a connected finite 'etale cover.

Now take $D = 26$. $X^{26}_0(1)$ is of genus 2 and hence is hyperelliptic. Take $N$ large enough and you're done.

I also guess there are easier examples but this was the first one to come to me.

Certainly not. Ogg proved in "Real points on Shimura curves" that there are only finitely many Shimura curves (of the form $X^D_0(N)$) which are hyperelliptic. If there is a prime divisor $q$ of $D$ such that $q\equiv 1\bmod 12$ then for all $N$ coprime to $D$, $X^D_0(N) \to X^D_0(1)$ is a connected finite 'etale cover.

Now take $D = 26$. $X^{26}_0(1)$ is of genus 2 and hence is hyperelliptic. Take $N$ large enough and you're done.

I also guess there are easier examples but this was the first one to come to me.

Certainly not. Ogg proved in "Real points on Shimura curves" that there are only finitely many Shimura curves (of the form $X^D_0(N)$) which are hyperelliptic. If there is a prime divisor $q$ of $D$ such that $q\equiv 1\bmod 12$ then for all $N$ coprime to $D$, $X^D_0(N) \to X^D_0(1)$ is a connected finite 'etale cover.

Now take $D = 26$. $X^{26}_0(1)$ is of genus 2 and hence is hyperelliptic. Take $N$ large enough and you're done.

I also guess there are easier examples but this was the first one to come to me.

edit: This is a counterexample for your second question as well because the cover in question is 'etale but not Galois in general (say for $N$ large enough). If the whole cover $Y \to X \to \mathbb{P}^1$ were Galois, then we'd have $Y \to X$ be a Galois cover as well.