I would say generally that the core principle should be the 'Non-Aggression Theory' or however it's put. People have rights. They cannot, however, exercise those rights to a degree that directly harms others and deprives others of their own rights. That should be the central framework. I also think that we should keep in mind the observation, held true without history, that the best government is usually the most limited and small government: like the force of fire, government power is useful as all heck but must be strictly contained.
To pick one particular point and chew on it:>"Should a government employee be allowed to not perform critical functions of their job?"
I'd say that this question hinges entirely on whether or not the employee is willing to step aside and let somebody else do the essential work. If somebody is being like Kim Davis and using their position to forbid anyone whatsoever from doing an important function, doing the equivalent of spreading one's body in front of a schoolhouse door lest it be racially integrated, then that can't be allowed. In those circumstances, real harm is happening. Other folks could do that work.
To put even more fine a point on it, suppose a police officer who's a Christian fundamentalist refuses to investigate a case of a transgender woman being raped and murdered because he just can't stand those individuals. That's disgusting. He ought to be fired immediately.
But what if we don't know what this hypothetical Officer Smith is thinking, and all that's happening is that we're in a police station where assignments are getting handed out semi-randomly. Smith gets a certain Jane Doe situation and says "Maybe Officer Roberts should handle this instead since he's in tight with the local LGBT community more than I am, with more experience." Even if Officer Smith's own religion happened to play a role in what he just said, even subconsciously, I'd say that no harm, no foul. Smith is being reasonable.