Neuro has pointed out a problem of using the centre point and then recomposing. It stuck in my mind, and here is a simple explanation. A lens is designed so that everything in a plane perpendicular to the lens is in focus. So, focus on the centre of a brick wall in front of you (flat facing you, of course), and all the bricks are in focus, including the ones at the edges, which are actually farther away. Focus on a brick at the edge, hold the focus and point to the middle, then the focus will actually be behind the wall because the bricks at the end were farther away when you focussed. So, if you focus and recompose, the subject might not be in good focus even if it is in the same plane.
Regarding 1Dx and 5DIII, I thought that they had the same AF system, which was probably the biggest plus of the 5DIII.