So we don’t actually need permutation polynomials here. In PLONK, we are trying to enforce that a set of wires is a specific permutation of another set of wires (or of itself). Here, we are just trying to enforce that the new commitment is any permutation of the old commitment plus the changes. The latter is actually easier to do than the former; in fact, the “specific permutation” argument is built out of the “any permutation” argument.
The keys are indeed not necessarily copied consecutively because there could be gaps.
The polynomial p for which you are applying p(x+1) = p(x) * .... is not the polynomial that we’re proving permutations of, it’s the “accumulator” polynomial that’s used to prove the permutation argument.