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...So based on what I read on the manual if I'm going to shoot at below 2.8 should I should I always stay with the center focus points? ... if I decide on using other than the center focus points should I be between F2.8-F4 since these are the only aperture these focus points are cross type?
What this means is that certain AF points will only work as cross-type or at highest accuracy when used with a lens of a maximum aperture of at least a certain value (2.8, 4 in the case of the 5D III). You can always use lenses of wider max aperture such as your f/1.2. Also, the shooting aperture does not matter. You can shoot the 50/1.2 at f/11 and get full functionality and accuracy from all AF points.
I find the off-center AF points in the 5D III very accurate. Unless there is a factory misalignment of the AF sensors, you should get good results from all of the cross points if the center point(s) are accurate. MicroAdjustment may or may not be necessary.
I find that for close range hand held work at maximum aperture, it helps to use servo AF even for still subjects. The reason is that I am not a human tripod, and can easily rock forward and back enough to put a subject outside the DOF of a fraction of an inch. You may want to experiment with that for things like portraits from 3 to 5 feet. Back button servo AF is nice for more than just action shooting!