A: Those are not considered “Actions” in the sense that they are not Action cards, and Stop That (or Belay That) is intended to cancel out Action cards specifically. Nor will Veto! which cancels Rules stop this type of free action.
More broadly worded Surprises might prevent some of these, however. For example, Skullduggery is designed so that it can prevent Plundering (among other things), which is a “free action” on a Rule. It’s A Trap and You Can’t Take This Guy From Me are designed so that they can prevent special Keeper actions that let someone steal one of your Keepers.
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 we would rule that it’s no longer stoppable by the Stop That! Anti-Action Surprise.