If the HU keeps turning itself on and off then I wonder if maybe some mis-wiring is upsetting the amp which then turns the HU on and off. Given that the HU works perfectly if you disconnect the speakers, I'd maybe try connecting the tweeters and woofers etc one at a time until you can work out exactly what is causing the problem.
Unless I'm mis-understanding you, I think you're under-estimating the complexity of a cross-over. The watts and ohms of the factory speakers are almost irrelevant to the cross-over (OK, it'll effect the power rating but not the values of the cross-over components). What does matter is the frequency at which the woofer and tweeter (and mid-range?) need to be crossed over, which will determine the values of the inductors and capacitors.
I see from your edit that there is possibly something to do with shared grounds and I reckon that your time would probably be better spent investigating this rather than trying to make a cross-over from scratch. (I have built amplifiers, loudspeakers and CD players in the past and I would always try to avoid re-building a cross-over from scratch.)
Anyway, good luck. Let me know if you have any queries.