You can check continuity between the fuse or skip to the relay, between it and pump connector with a multimeter. Add an extension wire if the meter probe leads are too short. Maybe a rodent chewed through the wire.
As far as your concerns about the dark green wire between the relay box and the connector, you can check continuity at any point with a multimeter probe, insulation piercing needle tip.
You can diy make a crude needle tip literally, by just taking a needle and binding it to the multimeter probe with a few turns of small bare wire. Suppose you have continuity between the relay box and the wire as it goes through the firewall, then you don't need to run a new wire through the firewall... if the wire is broken under-hood you can just resolder or add new wire to replace the bad piece instead of the whole wire run.
You can check continuity between every leg of the circuit...
- Battery to fuse 12 (with key in accessory or run position)
- Fuse 12 to front washer relay pin 5 and pin 1
- Relay pin 5 to pin 3 (with key in accessory or run position and washer column switch held on)
- Relay pin 3 to washer motor
If you do not get continuity between relay pin 5 and 3, but the rest checks okay, when washer should be on, measure for voltage between relay pin 2 and ground. If it is 12V, the GEM module is not activating the relay coil which should pull it down close to ground.
You can also manually, ground relay pin 2 to see if that allows the washer to come on. If it does then check continuity from relay pin2 to GEM pin 24 (same tan/red wire).
I know you have already done some of this, but I'm putting it all down for completeness in that order.
Edit: You posted this in the 2nd gen forum. If there are any differences in your '03, you might want to have the topic moved to 3rd gen.