I think no mods should be allowed to go on invisible mode. The second they are allowed to be anonymous is the second they can abuse their power and give out bogus warnings. It's the same reason why police get in trouble when they take off their badge/nametag.
No. I prefer them to be visible. When you try to throw shade and you've seen their name pop up in the currently browsing you run the **** out like avoiding some Aunt's kisses and "how's life doing?" during Christmas reunion.