Can NEW RULE cards that offer a "Free Action" as written on it, e.g. NEW RULE:Let's Keep Doing That, NEW RULE:Recycling, etc. have their effect cancelled by the SURPRISE:Stop That Action card?
See picture for an example.
I would say no because its not really and action at that time. But that is tricky
No. Those are not considered "Actions" in the sense that they are not Action cards, and that Surprise is intended to cancel out Action cards specifically. More broadly worded Surprises might prevent some of these, however. For example, Skullduggery is designed so that it can prevent Plundering, which is a "free action" on a Rule.
There might be some confusion on Let's Keep Doing That, since there is an Action card permanently in play, but it is intended to act as if it were a New Rule, so I would say that it's no longer stoppable by the "Stop That!" Anti-Action Surprise.