Since we don't have a parliamentary system, it's not possible to have a government not form at all. We can, however, have Congress deadlocked between the two parties, or an impasse between Congress and the President.
As to state secrets--I figured that much more would be a state secret on Cardassia. Some of this, of course, would be a good thing (sometimes our press goes over the line here), but some could be very bad, and really hide reality from people.
As for people doing an about-face, saying and doing one thing in one position, and then doing another when they get a promotion--it's not hard to imagine someone doing that. In fact, I still think that's how the True Way might climb back to the top, or other groups with similar intentions. If they are willing to be patient...
(BTW, it's funny that you came to the idea of a 10-year term...that was actually a number I had settled on for certain posts in the the SigCat