You don't need to have the engine running or even the ignition switched on. No, the relay doesn't switch a ground, the ground is there permanently on the black wire to the compressor. Power comes from one or other of the fuses and arrives at pin 3 on the relay (Relay 20), pin 2 has a permanent ground on it so a 12V supply from the ECU on pin 1 of the relay pulls it in and the power on pin 3 then goes out from pin 5 to the green wire on the compressor.making it run. So if you jumper pins 3 to 5, it will run unless it is dead. If it gets too hot, the thermal fuse opens which removes a ground to the ECU telling it that the compresor has overheated so it no longer energises the relay. If you have continuity between orange and black and the relay isn't being energised by the ECU, it is something else that has caused the ECU to inhibit any activity.
This is where you need to put the jumper
The ride heights are set according to the readings given by each height sensor. If you replace one then the chances of the reading being identical to the one you've taken off are pretty remote. It'll be similar but not the same. You need the software to be able to read and write the heights. I changed one of mine recently so before removing it checked the reading it was giving for the height that corner was sitting at which, in my case, was 130. With a jack under the body as well as the one under the axle so the two remained the same distance from each other, I then fitted the new sensor and checked the reading from that. This one was showing 135, so I added 5 to the stored settings on each height for that corner and wrote the new settings into the ECU. If I hadn't done that, it would have sat with that one corner permanently lower than the other three. Ideally it should be calibrated with a set of blocks but I know I've seen them somewhere since moving house but I'm buggered if I could find them. It's sitting level at all heights so it isn't far out but could do with being done properly when I find the blocks.