1. Applying a stiffness to the axial direction - x - with a gap will instruct caesar to ignore all stiffness until the gap condition is met, after which the stiffness is what you specify. After the gap condition is met, the stiffness now acts like a spring, but specifically not like friction. It'll spring back.

My interpretation of the design is if you blinded off either end and pressurized it, the expansion joint would provide almost no resistance to the pressure thrust up until the slot condition is met. Depressuring it will do nothing until it pulls a vacuum, after which it'll provide almost no resistance to contracting.

I would expect that from an expansion joint loading perspective, using unlimited stiffness after the slot condition is met is conservative.
I would expect from a nozzle loading perspective, using unlimited stiffness is non-conservative, but reasonably accurate.

2. Unless I misunderstood something and the gimbal is somehow a fixed point in space, I don't see it needing special consideration. My understanding of the mechanism is the gimbal is fixed to the center of the expansion joint, but that doesn't factor into the expansion joint simulation in CAESAR.

If the gimbal IS a fixed point in space (and not just a fixed point relative to the inner cylinder), please advise and we'll work from there.

In CAESAR, and expansion joint has forces that come out either end, which expands/contracts the expansion joint according to its axial stiffness, minus the amount that gets subtracted out the ends due to the stiffness of elements beyond nodes 10 and 20. Since we're assuming there is no axial stiffness to the expansion joint, it'll be up to the pipe to decide if P*A will move it and the slot conditions are met.