For starters, this is a fascinating problem. Reminds me of the Collatz Conjecture. I've been trying to formalize your hypothesis and achieve a result, but it is difficult. To review, you have asked three questions that are extremely relevant:
1) For N even, do there exist any "loops" or infinite cycles?
2) For N odd, do there exist any loops other than (1,1,1,...,1,0)?
3) Does the system terminate after at most 2^N-1?
The only solid result I was able to prove is that the range cannot increase. Consider starting with . Define Range(a)=max(a)-min(a). It is easy to see that max(b)<=Range(a). Since min(b)>=0, Range(b)<=Range(a). Therefore the range must be decreasing or constant. The questions can then be rephrased...
1) For N even, are there any sequences whose range stays constant?
2) For N odd, is (1,1,1,...,1,0) the only sequence whose range stays constant?
3) Can Range(a) decrease to 0 in more than 2^N-1 steps?
It's not much, but it's a step in the right direction, I think.