Changing AVL's balance factor to some other $s>2 \in \mathbb{N}$
If the maximal branch is of height $h$ then you know that all the other branches have at least height $h-s$. 

Lets consider $n$ the number of nodes in the two extreme case:


All branches of height $h$: $n=2^h$
One branch of height $h$ the others of height $h-s$: then $n=2^{h-s}+s$


Hence $ 2^{h-s}+s=\frac{2^h}{2^s}\leq n\leq 2^h$ since $s$ is a constant the height is still logarithmic in the number of nodes.

I think this answer your other questions too.