My 95 chevy 1500 pickup is having fuel pump issues. Somewhere between the relay and the sending unit. Pulled the tank, and put power directly to both the pump and the sending unit. They both run. Put in new relay, still not turning on the pump. Do not know if this is related, but the switch to turn on the interior light when you open the door does not work on the drivers side door. If I pull on the switch the light will come on.
It could be that the power to the relay has been disconnected, either a blown fuse, or a wire or connector is broken. It wouldn’t be unusual for a fuel pump connector somewhere to get fried. The pump pulls a lot of current. When that happens, it may look to be connected, but the connection is high resistance due to carbon all over the connections. First thing to do, measure the voltage at the pump, when it is supposed to be running. Back probe the connector closest to the pump, so that you are measuring the pump voltage with everything connected. Does it have power? If not, then you’ll have to work backwards from there. You’ll find this much easier if you can secure a wiring schematic. This isn’t likely to be related to the interior light problem. Esp since it comes on when you jiggle the switch.
Be aware that if the problem turns out to be a fried electrical connector in the pump circuit, that might mean the pump - even though it seems to work – is bad and pulling too much current.
You need at least a test light probe to make the checks for power and follow the good advice given by @GeorgeSanJose. You just need to find where the bad connection to power is at. The light will show you where it is and where it isn’t getting to.
The door switch problem is with the switch itself. The spring action in the switch isn’t pushing on it hard enough to make the ground connection to turn the light on. Spraying some WD-40 on it might help getting it to work again.
You can’t determine the pump is good because it ran. It could be tossing fuel to the side via bad hose or simply not pushing enough.