My original answer missed the word "span" everywhere, sorry!
The answer to your second question is no. Take $$A=\begin{pmatrix}0 & 0 & 0\\ 1& 0 & 1\\ 0& 0 & 0\end{pmatrix}, B=\begin{pmatrix}0 & 0 & 1\\ 1 & 0 & 1\\ 1 & 1& 1\end{pmatrix}.$$
If I computed things correctly, the dimensions of the spans go $2,5,8,9$. Thus $N(3)\geq 4$. (This example was found by taking random $0,1$ matrices, and calculating dimensions, until an example was found.)