[blue62]

Luckyman,

Looking at the wiring diagram for the fuel pump relay you have four wires coming into the relay plugin.
They are labeled on the relay as..
30
85
86
87...

30 is the path to the fuse.... it is switched power to the relay.
when tested with a volt meter it only shows voltage with key on.

85 is ground at the ECU.. always shows ground key in key out... on or off.

86 is constant battery voltage..key in key out on or off.

87 is voltage supply to the fuel pump when the relay contacts close.

you know 30 and 87 are good because you jumped those and the car runs.

So test 86 and see if it shows constant battery voltage.
When you test 86 record the voltage reading you get.

also test 85 and see if you have constant ground.

Also test 30 and make sure you have voltage with the key on.
Record that voltage

Let us know the results.

Forget about the DME relay or other issues for now.
Chasing more then one issue at a time usually causes more problems then fixes.