If it is just a regular relay, then it's easy to check it, just get a small 9V battery, and connect it to the 2 poles of the relay which are supposed to be connected to the control, and then the coil should work and the relay should click, the two other poles should be then get connected together (there shouldn't be any resistance between them). When you disconnect the battery
the two poles should be disconnected too. If it is more complicated relay, it might let's say connect two poles for only several seconds when you turn it on, or something else(depending on the circuit that's in there).
What about the Z, I think you better find out what's causing to get it fried ASAP, until something else, more expensive or more "hard to rich" won't get fried...