In general, for $n\times n$ matrices $A,B$, the matrix product $AB$ represents the composite map obtained by applying $B$ first then applying $A$. This is simply because for any vector $v$, $ABv=A(Bv)$, so multiplication by $B$ is done first.
I believe that $1$ describes $M$ rotated about my space's $Z$-axis and $2$ describes rotating $M$ about it's $Z$-axis.
No, this is not correct the way it is written. What does it even mean for a matrix to be "rotated" about an axis? It is however true that $MR_z$ corresponds to the linear transformation which first rotates the space about the $z$-axis, and then applies the linear transformation defiend by $M$. On the other hand, $R_zM$ corresponds to applying $M$ first, and then rotating about the $z$-axis. In general, these two are not the same, of course.
It would also be correct to say that $R_zM$ has column vectors which are rotated copies of the column vectors of $M$. That is, if
Of course, you will observe that $R_zv_i$ is just $R_z$ applied to $v_i$, which corresponds to $v_i$ rotated about the $z$-axis (for each $i=1,2,3$). Furthermore, $MR_z$ has row vectors which are rotated copies of the row vectors of $M$. That is, if
Of course, you will again recognise that $u_iR_z$ is just a rotated copy of the $u_i$, for each $i=1,2,3$.