Compound Node–Kayles on paths

In his celebrated book [J.H. Conway, On Numbers and Games, Academic Press, New-York, 1976, Second edition (2001), A.K. Peters, Wellesley, MA], J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node–Kayles game on paths. For usual disjunctive compound, Node–Kayles has been solved for a long time under normal play, while it is still unsolved under misère play. We thus focus on the ten remaining versions, leaving only one of them unsolved.