[LS short box]

So the ECM needs 12 volt constant (all the time) and 12 volts in start and run.
The purple wire is the keyed start wire to the starter. If you are running an auto you need to run the purple wire to the neutral safety switch and then to the starter.
Yes on the fuel pump wire. That wire should be part of your LS engine harness relay. When you turn the key the pump should run for a few seconds if you don't go all the way to start. To keep the fuel pump running the ECM needs to see crank shaft rotation.