Live view focus works by adjusting the lens until the image looks sharp on the image sensor. Since that's the same sensor as registers your pictures, if one is sharp, the other is too.
The normal AF system works with a separate sensor, hence bad alignment of that sensor may cause out of focus images. The same can happen if the lens isn't properly calibrated.
Focusing works the same in P mode as in Av or M or whatever. Only preprogrammed modes, like the green box, are different.
All focusing is done with the aperture wide open, so live view focus differs only if the max opening is smaller than f/5.6. It usually is not, but may be, if you have extenders on slow lenses.
Microadjustment can only change one thing for a lens. A zoom lens calibrated at a repair center is adjusted at eight different points. Don't expect micro adjustment to solve your problem. If it does, it often solves it at a certain focal length, a certain distance to the subject and sometimes even under certain lighting conditions. In other conditions, it may even make things worse.
But if it does solve one problem, it will at least indicate if your lens/camera needs calibration.