2006 Honda Civic R/S Problem

I've never used the omega unit, but as far as the dei unit goes, no you don't need the pink wire.  Before you assume it's the bypass module causing the problem, disconnect the bypass module.  Take the key and put it into the ignition, don't turn it or anything.  Then try remote starting the car a couple times.  IF the car starts each time then your bypass is at fault.   I have in the past had problems with the 556u not activating properly.  The way I've found works is to take the both the black and the blue wires and hook them up to the ground out while running of the r/s.  Hopefully this helps.