How to compute powers of square matrices?
As you note, computing $A^m$ can be done in $O(\log m)$ times the number of operations for matrix multiplication on $N \times N$ matrices. The answer to your second question is no, at least for asymptotic complexity -- matrix squaring and matrix multiplication have equivalent time/arithmetic complexity (up to constant factors). Reducing squaring to matrix multiplication is obvious. To reduce multiplication to squaring, suppose we wish to compute the product of $A$ and $B$. Form the $2n \times 2n$ matrix $C$ with the block structure:

$[0~~A]$

$[B~~0]$

That is, $C$ has an $n \times n$ all-zeroes matrix in its upper-left quadrant and lower-right quadrant. Note that $C^2$ contains $A\cdot B$ in its upper-left quadrant.