I would check the fuel pressure at the fuel rail behind the engine head. The other thing that comes to mind would be that the idle air control valve is closing down, or that you have a vacuum leak.
I -think- that on these Jettas, the Idle Air Control valve goes to 'completely shut' when the engine is warm. The actual idle speed then being controlled by an adjusting screw that keeps the throttle valve slightly open. (This is different from most American fuel injection systems, where the throtle valve is fully closed when you foot is off the gas - and the IAC does the controlling of idle speed.) It also means that every 100,000 miles you may need to adjust the idle speed, as the adjusting screw wears down against the throttle mechanism.
I have a 90 Jetta (also 1.8 engine & Wolfsburg Ed.) that runs fine, though the IAC is -absent- and the two hoses are plugged shut. Its a little hard to get started on a cold day, but since its in South Texas, not usually a problem. A malfunctioning IAC could explain the wide excursions in idle speed.
But a bad vacuum leak could create your symptoms. Make sure the rubber duct coming from the air filter housing doesnt have rips/tears in it. The computer needs to see the input from the mass air flow sensor in that air filter housing, to decide to run the injectors the right amount. Its the only way the computer knows how much air is coming in.
I measured the fuel pressure on mine with a home-made tool. There is a little bolt in the very end of the fuel rail, on the end toward the drivers side. If you unscrew that bolt, a little fuel should squirt out - so dont be smoking when you do this. (be sure to keep the housing from turning if the tiny bolt is tight). I think its an 8 mm bolt, or maybe a 7 mm. I clamped a length of 5/16" hose over the end of that little tube, and put a pressure gauge on the other end of the hose. Any guage that will read above 50 psi will do. I used an antique one from an old hot water boiler. With the key turned on (more than once, to build up the pressure) but without starting it, it should show something like 40-42 psi - if I recall correctly. With the engine running it will drop back to perhaps 32 psi at an idle, and somewhat higher pressures (like 36-38 psi) when you push the gas pedal down suddenly. That is because the fuel pressure regulator controls fuel pressure based on how much vacuum there is in the manifold. If you see pressures lower than these by more than 20%, either you have a fuel problem - or your gauge is way off.
Your car has 2 fuel pumps. One is under the car, on the passenger side below the back seat. The other one is inside the tank. My vague impression is that the car will sort-of-run if the pump in the tank has died, but wont start and run if the 'main' fuel pump has quit. You should be able to hear them run for about a second after the ignition sw is turned on - if its a quiet location. There is a fuel filter next to that main pump - though I havent taken mine apart yet. (knock on wood). By listening at the fuel filler neck, you might be able to tell if the in-tank pump is running.
I havent had any issues with the fuel pump relay on my Jetta - but the Volvo 240's from the later 80's had the same type of fuel pump system, and had a fuel pump relay that was known for going bad. Unfortunately I dont know which of the many relays on the Jetta is for the fuel pump.
If you need to check the fuel filter -sock- inside the fuel tank, you access the fuel pump/sender/pickup assembly through a hole in the floor of the trunk. Rarely that sock can get clogged. Though not with rust, the Jetta fuel tank is ABS plastic.
Dark or black plugs can result from it running too rich - or from running so lean that it misfires a lot, or from running with the spark retarded. And once they are -black- they can be fouled out, and not pass the spark when trying to start it. I dont know if the Jetta distributor has a vacuum advance mechanism in it - if so, make sure that is working freely. If its gets stuck and doesnt advance, your plugs will get pretty dark, and you will have Feeble power at driving speeds, if it will even get above 40 mph.
Let us know how it turns out. Then I can learn too.