I'm having troubles finding a list over the most basic arithmetic rules of matrices. For example, in a proof using a chain of equivalences, can I choose exactly where to multiply a matrix? Like this:
$$ AB=BA \Longleftrightarrow CAB=CBA \Longleftrightarrow ABC=BAC \Longleftrightarrow ACB=BCA $$
Are all of these equivalences OK? Or maybe only the first and the second one?
That chain of equivalences is correct, for the most part. What you can say is this $$AB=BA\Leftrightarrow CAB=CBA\\ AB=BA \Leftrightarrow ABC=BAC$$ with $C\neq 0$ and, as the user cdwe pointed out, only if $C$ is invertible (see the counterexample in the comments). If you say $$CAB=CBA\Leftrightarrow ABC=BAC$$ without implying that $A$ and $B$ commute, is not always true, because what you're saying is that $C[A,B]=0$ is the same thing as $[A,B]C=0$ and this is only true if the matrix $C$ and the matrix $[A,B]$ commute with one another or that $[C,[A,B]]=0$ (or $C$ could simply be the null matrix, but that's a special case because the zero matrix commutes with every matrix).
So to put it in a better way
On a side note, you have that if $[C,[A,B]]=0$, then $[A,[B,C]]+[B,[C,A]]=0$ directly from the Jacobi identity that says $$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$$
The third equivalence is not right at all: you could either multiply on the left or multiply on the right, there's no such rule for multiplication "in the middle", just take as an example $A=I$ and $$B=\left(\begin{matrix}0&1\\0&0\end{matrix}\right) \\ C=\left(\begin{matrix}1&1\\1&1\end{matrix}\right)$$ you can easily check that $A$ and $B$ commute but $ACB \neq BCA$