I suppose that, as you say, they are worried about adding fuel to the fire by firing the parties who were actually at fault. Personally, I doubt it would. What's most likely to happen is that they get it wrong in both cases: first, damaging their reputation by firing someone who didn't do anything, and then later not firing someone who did damage their reputation, thus damaging their public reputation (perhaps) and hurting their internal reputation and morale (one would assume).
It will. In this example, publicly firing the guy who ordered Sleiman's termination can be construed as "American firing Hamas Sympathizer fired and sympathizer reinstated"
A culture war is a two way street. No reason to further fan the flames.
It’s like the difference between a salesman ranting on Facebook about how a competitor sucks, versus a marketing campaign that’s an attack ad on a competitor.