I also had MAF issues that made the car run like crap, knock, and retard timing. New (not reman) Denso part fixed it.
Sounds like you may have a short. Might be the result of a shoddy turbo replacement job: disconnected/damaged vac lines, could have nicked the wiring harness if not run in original locations, etc. I would test the alternator and the battery before diving into tracking down a short, though. A battery that is down to 4.x volts is basically perma-dead and needs to be replaced, no matter how many times you try to charge it.