Well as I guess you are aware it is most likely an electrical problem. Do the easy stuff first - cleanliness of battery terminals, clean up all earth points and protect with petroleum jelly. If the car at some time or other used to work with out these problems I wouldn't be ripping stuff out just yet - especially with a bypassed alarm (sounds like a nightmare)...

Look also for electrical connections that are close to heat sources - I once owned an old V6 Ford that had an electrical plug that rested on the top of the engine... the engine got warm - the contacts broke - you'd just hear the air whistling by and then a loud bang as the engine mis-fired and started again. Great.