2D Rubik's cube?

If we start with a matrix

$$\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$

then the map $1 \mapsto a$, $2 \mapsto b$, etc, defines a permutation $\sigma$ of $\{1, 2, ..., 9\}$, i.e. an element of $S_9$. The valid "moves" consist of composing this permutation with one of the elements $(123), (456), (789), (147), (258), (369)$ or their inverses.

So the question becomes: what is the subgroup $G$ of $S_9$ generated by $\{(123), (456), (789), (147), (258), (369)\}$? For starters, these are all even permutations, so we're in fact looking for a subgroup of $A_9$.

Next we notice that we can get from $A$ to

$$\begin{bmatrix}2 & 3 & 1 \\ 5 & 6 & 4 \\ 8 & 9 & 7\end{bmatrix}$$

and then to

$$\begin{bmatrix}2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 1\end{bmatrix}$$

so $(123456789) \in G$. And $(123456789)$ and $(123)$ together generate $A_9$, so $G = A_9$.

So the answer to the first question: you can get from a matrix to $A$ if and only if the permutation $\sigma$ defined above is an even permutation.