If I have Rule 1 run the actions of Rule 2, will the required expression of Rule 2 have to be satisfied for Rule 2 actions to run?
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No. "Run Rule Actions" simply runs the actions of the specified rule and will do so regardless of any Required Expression. [EDIT: Except as described below.]
Not quite. If the Required Expression has the option to Cancel Pending Actions when false, the actions of the called rule will not run. There is a bug in so far as when this happens, there is no log entry provided to say that this is what happened (well, didn't happen).