I have a 1985 RX-7 with a fuel pump relay that is always open.
I took it off and checked it; with no power to it, it is closed. When I reinstalled it and turned the key on it opens.

This means that my fuel pump won't pump because it is getting no power.

I jumpered it like the service manuals say to do when testing the fuel pump.

What controls the fuel pump cut relay and what purpose does it serve?

 

The relay is controlled by the ECU. It should be at ground on the Black/Yellow (relay to ECU) with the key on. My Mazda manual doesn't mention any condition where it should not be at ground. Looking at my 13b version ('84), I can't see why it should ever be open: It has a relay with two coils ( N/O contacts). One coil will activate the relay with the ignition in "start", the other coil takes over in "run" as it is grounded by a switch in the air flow meter, activated by the flap. So the contacts are always closed. Logically, yours yours would be also.
Grounds? The ground for the ECU is not, according to my diagram, a tab near it. It is the general engine ground. Apart from the two obvious ground points on the battery cable (left strut tower and engine), the diagram shows one under the airbox, and it shows one under the dash, above the pedals for the -SE but not for the 12a. Might be worth looking under there in case the book has an error.
Weak connector at ECU?
Bad solder joint in ECU?
Hope it's not a dead ECU.
I'd be interested to know what you find.

-John.