I found a solution trying to minimize prime factors. And finding a balance between the minimum value and minimum number of factors.
I noticed that some fields are connected, in that multiplying one of them results in the multiplication of fixed other fields. There are three such patterns:
-C, D, H, L, and M (and any mirror image of that)
-A, E, G, L, M (...
Edit: In an effort to find the minumum, here is a much smaller solution in which the mutual product is
As MKBakker pointed out we could further reduce this by dividing each of the entries 4,8,16,96 and 192 by 2 to get a mutual product of
although they have subsequently improved on this.
To formalize user61579's answer, a simple reinterpretation of the rules is that, defining a pass as a move from the $1$ all the way around the sequence, every pass, every second $-1$ becomes $0$
Then, it is trivial to show that the state of a length-n sequence after each pass corresponds to the first n columns of this infinite table:
1 -1 -1 -1 -1 -1 -1 -1 ...
For the first question, there is an easy pattern that we can detect. First, all of the nice jumps will come from the n-1 position for an array of length n.
As for how many, I'll print the first few and then try to point out the pattern
2 and 3 are basically special ...