ag.algebraic geometry - Quotient of an affine variety by an infinite discrete group

This is a pretty basic question, so I'd be happy with either standard references or with explanations. Also, there's a good chance I'm confused about some things in the statement of the question, and corrections of these would also be cool.

If G is a finite group acting on a (commutative) $ mathbb{C} $-algebra A, then we can define $Spec(A)/G$ as $Spec(A^G)$ (where $A^G$ is the invariant sublagebra of A), and there is an algebraic map $Spec(A) to Spec(A^G)$ whose "fibers" correspond to orbits of G on $Spec(A)$.

If L is a lattice in $ mathbb{C} $ then $ mathbb{C} / L$ as a topological space can be given the structure of a scheme (it's an elliptic curve). However, this scheme is not $Spec(mathbb{C}[x]^L)$, since the only polynomials invariant under the action of a lattice are the constant polynomials. Is there a construction which replaces $Spec(mathbb{C}[x]^L)$ in this case?

Now let's assume X is a smooth affine scheme and G is a countable group acting freely on X (in my case $G = SL_2(mathbb{Z})$, but I would guess this isn't too important). Since X is smooth we can also view X as a complex manifold $X^{an}$, and then the quotient $X^{an}/G$ is a topological space. Can X/G naturally be given the structure of a scheme? If the answer is no in general, are there conditions on the data X, G that ensure the answer is yes?