You may have more than one problem, so lets fix
the most important one first.
The fuel pump runs when the key is turned from "Off" to "Run" with the engine not running.
By design AND for safety reasons the fuel pump control system was engineered so that the fuel pump WOULD NOT RUN when the engine was not running. Consider this, you are driving down the road and the fuel line breaks. The engine will run out of fuel and shut down. If the fuel pump is still running you will pump fuel out of the leak. This may not be important if the only thing you do is blow up your Zed and yourself. What happens if you are in a collision with a mini-van loaded with the midget hockey team?
The relay control system for 1978 and later has three relays: the E.F.I. Main Relay, the Fuel Pump Control Relay and the Fuel Pump Relay. Current is supplied to the injectors and the ECU anytime the ignition is in the "ON" position. BUT, the Fuel Pump and Auxiliary Air Regulator will operate ONLY if the engine is RUNNING or BEING CRANKED. The way in which "engine running" is sensed is by monitoring the engine oil pressure and the alternator output. If BOTH oil pressure and the alternator output are lost, the Fuel Pump and Auxiliary Air Regulator will no longer receive current, even if the key remains in the "ON" position. If only one is lost the Fuel Pump and Auxiliary Air Regulator will continue to operate.
With the key in the "Start" position, the Ignition Relay activates the EFI Main Relay to send power to the injectors and the ECU. Current is also sent to the Fuel Pump Control Relay. The "S" terminal of the Ignition Switch also sends current to the Fuel Pump Control Relay "S" terminal and it energise the Fuel Pump Relay. Thus, during starting, the Fuel Pump will run and the Aux. Air Regulator will get power even though there is no Alternator Output or Oil Pressure.
With the key in the "ON" position and the ENGINE RUNNING current flows through the E.F.I. Main Relay to the injectors and the ECU. Current also flows to the Fuel Pump Control relay, which merely passes it on the Fuel Pump relay. Thus the Fuel Pump and the Auxiliary Air Regulator continue to operate.
In the event of an alternator failure, one set of the windings in the Fuel Pump Control relay will be grounded at the alternator "L" terminal. This closes the corresponding contacts, BUT since the oil pressure switch is still open, the second set of contacts in the Fuel Pump Control relay WILL NOT be energised. The Fuel Pump relay will continue to receive current, and the Fuel Pump will continue to operate.
On the other hand, if the engine oil pressure drops, the oil pressure switch contacts will close. But, since the alternator output is normal, there will be no additional current flow through the Fuel Pump Control relay and the Fuel Pump will continue to operate.
Now, if the engine stops due to a severed fuel line, or simply dies during warm-up, the alternator output will drop to zero. This grounds the first set of windings in the Fuel Pump Control relay, closing the corresponding contacts. Since the Oil Pressure Switch contact will close due to the loss of oil pressure, current can now flow through the second set of windings. This causes the double contact arm, in the Fuel Pump Control relay, to be pulled down to the Fuel Pump Control relay "S" terminal. Since there is NO CURRENT at this "S" terminal UNLESS the key is in the "Start" position, current flow through the Fuel Pump relay windings will be interrupted, the contacts WILL OPEN, and the Fuel Pump will SHUT DOWN.
Mentioned several times were the "Alternator "L" terminal" and the "Oil Pressure Switch".
The Alternator "L" terminal is a transistor in the alternator. When the engine is not running with the key in the "Run" position this transistor is closed to ground. This is connected to the Idiot Light in the volt meter and because the transistor is conducting to ground the light is on. This transistor also provides a connection and a ground path for the first set of windings in the Fuel Pump Control relay.
The Oil Pressure Switch is in the Oil Pressure Sending Unit. It has two wires connected to it. The Yellow / Green wire goes the Oil Pressure Gauge to indicate the engine oil pressure when the engine is running. The Brown wire goes to the Oil Pressure Switch. This switch is CLOSED when there is no oil pressure. It provides a ground path for the second set of windings in the Fuel Pump Control relay. It also provides a ground path for the Oil Idiot Light in the dash Oil Pressure Gauge.
Now having said all of this there is a design flaw in the whole thing.
If the alternator "L" terminal is conducting to ground (charging Idiot Light ON) AND the Oil Pressure Switch is open (oil Idiot Light OFF) the Fuel Pump will RUN when the key is switched from "OFF" to "Run" even if the engine is not running.
If the alternator "L" terminal is conducting to ground (charging Idiot Light ON) AND the Oil Pressure Switch is closed (oil Idiot Light ON) BUT there is NO CONNECTION to ground at the Brown wire at the Fuel Pump Control relay, the Fuel Pump will RUN when the key is switched from "OFF" to "Run" even if the engine is not running.
Both of the above could be caused by several things such as, defective Oil Pressure Switch (always open), defective Fuel Pump Control relay or a broken wire somewhere between the Oil Pressure Switch and the Fuel Pump Control relay.
This is the way it worked from 1978 through 1981 280Z, 280ZX N/A's, 810's and 200SX's when they were built and this is the way they are supposed to work today. PERIOD!
As far as your concern about the FPR hissing, at this point it is normal. Fix your Fuel Pump Always Running problem first. If the fuel pump is always running and the engine is not running no fuel will flow through the injectors and ALL the fuel will try and flow back to the fuel tank. Since the FPR return orifice is small the fuel pressure will be higher than "normal". "Normal", in this instance would be ~ 37 PSI, with 0 inches of vacuum at the FPR vacuum sense line. You are getting 40 PSI, so your pressure gauge could be out by 3 PSI. You report 35 PSI when the engine is running BUT you also MUST measure the amount of manifold vacuum that is reaching the vacuum sense line of the FPR at the same time. If your intake manifold vacuum was in the area of 10 to 15 inches then your fuel pressure would be in the range of 32 PSI. You have to measure it.
Suppose you decide not to TEST and VERIFY and change the FPR with a brand new one and end up with the same results. What is the next thing you would change? Even worse would be swapping in another FPR that was used BUT defective, where do you go from there.
Please, do yourself a favour, get a service manual before you drive yourself crazy and try to not follow inaccurate advice.