I have the following questions:

Suppose the compact Riemann surface $C$ is an n-fold branched cover of $\mathbb{P}^1$ branched at exactly four points $x_1,x_2,x_3$ and $x_4$. I believe that $C$ is a smooth genus $(n-1)$ Riemann surface. Is this true?

In case 1 is true, what is the algebraic equation for this Riemann surface? [Naively, I thought $y^n=(x-x_1)(x-x_2)(x-x_3)(x-x_4)$ would do the job, but this surface is singular when it is given as a sub-variety of $\mathbb{P}^2$. Do I have to embed the initial $\mathbb{P}^1$ in a more complicated weighted projective space? How do I resolve the singularity to get a smooth genus $(n-1)$ surface?]

Given the algebraic equation for the Riemann surface $C$ of question 2, what are the $(n-1)$ holomorphic differentials?

I have a basic knowledge of Riemann surfaces obtained through undergraduate/early graduate level courses, but would like to learn more. In particular I would like to be address problems similar to questions 1, 2, 3 on my own with relative ease. Do you have any references(on/offline) that I can learn from?

Thank you in advance!