The first thing that comes to mind is you wouldn't want to tell your skip level manager anything that you wouldn't (or haven't already) told your direct manager. That's going outside the chain of command and creates political and organizational issues that no one will appreciate.
For this is less about being careful, and more about not going off on a spontaneous rant, which honestly, isn't a good idea to do for the chain of command anyway. Spontaneous rants are for your peers. Deliberate communication up and down the COF.
My two cents anyway.
I'd refuse to work anywhere that cares about chain of command. Managers are for bookkeeping purposes, not commanding. If your manager doesn't let you talk to the people you need to talk to to get things done then it is a very toxic environment.
Of course this isn't a hard rule, if I'm having a conversation with my skip-level and some topic happens to come up that I haven't spoken with my manager about before, I'm not necessarily going to hold back just because of that, I can follow up with my manager later. But if something's bothering me or I have some feedback or something like that, I'd generally chat with my manager about it before anyone above them.
I always did whatever the higher power wanted. It served me well.
True, but say something that reflects negatively on your manager, and if they hear about it, they may rapidly become _more likely_ to fire you than the skip-level manager.
In that case you still need healthy communication with your own team manager, but you wouldn’t be telling them everything except what they need to know to manage your workload.
Not in my experience. For my manager to fire me I need to be first put on an improvement plan. Skip manager can waive this and fire me on the spot, risking a potential lawsuit for the company.