Let $G$ be a finite subgroup of automorphisms acting on an affine variety $X\subset\mathbb{A}^n$.
I'm trying to prove that the natural projection $\pi:X\to X/G$ is a finite map.
First, I'm a bit confused about the quotient $X/G$. If I've understood it correctly, according to Shafarevich's Basic Algebraic Geometry I, section I.$2.3$, example $11$, the quotient can be seen $X/G$ as an affine variety (since $A(X)^G$ is a finitely generated $k$-algebra). But take $X=Z(x^2+y^2-1)\subset\mathbb{A}^2$ and $G=\{id, \alpha\}$, where $\alpha$ is the antipodal map. Don't we get $X/G\simeq\mathbb{P}^1$, which is not affine?
Besides, how do I prove that $\pi^*:A(X)^G\hookrightarrow A(X)$ is integral? I can see it for specific examples (like the one above), but not in general. Any hints?