Yes the car will start and run without the drivers side crank sensor in (when I was trying to get a Zed started we were swapping sensors from another Zed and found this out)
I don't recommend it though as it won't be able to monitor VVT!
The reason it won't start with the passenger side crank sensor out is because it uses this one along with the crank position sensor for timing so if that one is out the coil packs won't fire (or at least not in time)
Faulty crank position sensors can give some random errors and cause rough idle and running as can a damaged flywheels (the keyway on the back is used by the crank sensor for timing) as this is part of the baseline timing for the engine.
This is all from personal experience that tracked back to a faulty crank sensor - this can be found on the front edge of the gearbox near the clutch slave cylinder (assuming a VQ35DE engine)