I have exchanged e-mail with some specialists on invariant theory and affine algebraic geometry, and here is a summary of what I learned.
For the sake of simplicity (although this is by no means necessary), I will assume the base field to be $\mathbb{C}$. Again, I will denote by $P_n$ the polynomial algebra in $n$-indeterminates.
My original question was, given an arbitrary (in general, non-linearizable) finite group $G$ of automorphisms of $P_n$, to find necessary or sufficient conditions for the subalgebra of invariants $P_n^G$ to be again polynomial.
In this generality, with current knowledge, an answer to this question is hopeless. The main reasons are that, for $n>2$, the structure of $\operatorname{Aut} P_n$ is to a large extent unknown, and that very few families of non-linearizable group actions are known.
However, there is something that can be said in this generality. Instead of focusing the atention on $P_n^G$, we can considerer separating invariants, introduced by Derksen and Kemper, which is a weaker concept than generating invariants.
Let $X$ be an affine variety (not necessarily irreducible), and $G$ be a finite group of automorphisms of it. A subset $S \subset \mathbb{C}[X]^G$ is called separating if, for all $x,y \in X$, the following holds: if there is an invariant $f \in \mathbb{C}[X]^G$ such that $f(x) \neq f(y)$ then there exists $h \in S$ such that $h(x) \neq x(y)$. $\gamma_{sep}$ denotes the smallest interger $m$ for which there exists a separating set of size $m$.
An automorphism $\sigma$ of $X$ is called a reflection if the fixed subspace $X^\sigma$ has codimension at most $1$. Then we have a result by F. Reimers (Theorem 2.4 in the paper Separating invariants of finite groups) that says
Theorem: Assume that the $G$-variety $X$ is connected, Cohen-Macauley and that $G$ is generated by elements with a fixed point. If $\gamma_{sep}=\operatorname{dim}X$, then $G$ is generated by reflections.
Remark: If $X=\mathbb{C}^n$ and $G<\operatorname{GL}_n(\mathbb{C})$, every element has the origin as a fixed point.
My second question was about affine varieties $X$ and finite groups of automorphism $G$ such that $X/G$ is isomorphic to $X$.
When $X$ is an algebraic torus, there is a nice result in this direction, which is a multiplicative invariant theory analogue of Chevalley-Shephard-Todd Theorem.
Let $G$ be a finite group. A $G$-lattice $L \simeq \mathbb{Z}^n$ is a lattice together with a faithful homomorphism $G \rightarrow \operatorname{GL}_n(\mathbb{Z})$. Then $G$ acts on the group algebra $\mathbb{C}[L]$, which is just the algebra of Laurient polynomials $\mathbb{C}[x_1^\pm, \ldots, x_n^\pm]$, which is of course the algebra of regular functions of $\mathbb{T}^n$.
An element $g \in G$ is called a pseudo-reflection if it acts on $\mathbb{Q} \otimes_\mathbb{Z} L$ be pseudo-reflections, and $G$ is called a pseudo-reflection group if it is generated by pseudo-reflections.
Then we have the following theorem (see Theorem 7.1.1 in M. Lorenz book Mutiplicative invariant theory)
Theorem: Let $L$ be a $G$-lattice, for finite $G$. Then the following are equivalent
a) $\mathbb{C}[L]^G$ is regular.
b) $\mathbb{C}[L]$ is a projective $\mathbb{C}[L]^G$-module.
c) $\mathbb{C}[L]^G \simeq \mathbb{C}[\mathbb{Z}_+^r \oplus \mathbb{Z}^s]$, a mixed Laurient polynomial algebra, with $r+s=\operatorname{rank}(L)$.
d) $G$ is a pseudo-reflection group and $\mathbb{Z}[L]^G$ is a unique factorization domain.