Let $G$ be an abelian group and let $R$ be a $G$-graded commutative ring, i.e., $R=\oplus_{g\in G} R_g$ with $R_gR_h\subseteq R_{g+h}$. Let $M$ be a $G$-graded $R$-module i.e. $M=\oplus_{g\in G}M_g$ and $R_g M_h\subseteq M_{g+h}$. We will say that a morphism of $G$-graded $R$-modules $f:M\rightarrow N$ has degree $h\in G$ if for all $g\in G$ one has that $f(M_g)\subseteq N_{g+h}$.

Q1: Is it always possible to find a resolution $F^{\cdot}$ of $M$ by Free $R$-modules which are $G$-graded and where the differentials have degree $0$ ?

Q2 If the answer to Q1 is no, then what are further assumptions that one can impose on $R$ and $G$ in order to guarantee the existence of such a resolution?

The motivation for $Q1$ and $Q2$ is this: If $M$ and $N$ are $G$-graded $R$-modules, is it possible to put a natural $G$-graded structure on $Tor_R^i(M,N)$?

Q3: What is a good reference for graded homological algebra?