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Well from the clues i read, i suspect that maybe your ignition switch is possibly worn out. Because after all, the power for almost everything in the car goes through it. Try jiggling the switch next time it acts up, and see if this affects anything.

This isnt as far fetched as it may seem, my Dad and I changed the engine on his truck once and it started and ran fine first time. Then the next time it wouldnt start at all! then it would start, then it wouldnt, etc. Finally we traced the problem to the worn ignition switch. And upon replacing that, it ran fine.