I believeMy original answer missed the word "span" everywhere, sorry!
The answer is that thereto your second question is no finite bound. See my similar question here: Take A back and forth Euclidean algorithm over the integers--does it have bounded length?$$A=\begin{pmatrix}0 & 0 & 0\\ 1& 0 & 1\\ 0& 0 & 0\end{pmatrix}, B=\begin{pmatrix}0 & 0 & 1\\ 1 & 0 & 1\\ 1 & 1& 1\end{pmatrix}.$$
Edited to add: Your argument aboutIf I computed things correctly, the dimensions of the spans go $N(d)=d^2$ fails because these sets are not subspaces$2,5,8,9$. Thus $N(3)\geq 4$. (This example was found by taking random $0,1$ matrices, and calculating dimensions, until an example was found.)