Here it is: In the new, official rules, can you use Running shot and RCE at the same time? How about Charge and CCE?

No and no. Running shot requires a Power Action and then allows the character to make a Ranged Combat Action as a Free Action. So you can only give the character a Ranged Combat Action to make the attack portion. RCE requires a Power Action. The same logic applies to Charge and CCE.

According to the Oct 1st FAQ, Charge and CCE are both power actions and cannot be used together. The same goes for Runnning Shot and RCE. This is specifically addressed in the FAQ.

They wouldn't do something that dumb, especially after the dominance that RCE had on game play.