2001 GMC Sonoma fuel pump doesn't run

There's nothing magic about one or two seconds. The point is it better turn off until you crank the engine. That's to insure the pump turns off if a fuel line is ruptured in a crash. The engine will stall because it can't run without fuel pressure. That's when the relay turns off so the pump doesn't dump raw fuel on the ground.

The most valuable clue here is your observation of 7 volts at one point. You most likely would have had 0 volts if you were using a test light. This sounds like a high-resistance / corroded connection. All you need is one tiny strand of wire for enough current to get through for the voltmeter to pick it up. My guess is you had the pump unplugged when you found that voltage. If it was plugged in, the motor would have loaded the circuit down and you'd have found 0 volts. The test light also loads the circuit down, though not as much. Not enough current can get through the break to run the test light.

I'd swap in a different relay first just in case the contacts are arced or pitted, but it would be more common to find a break in the wire between the relay and the pump.