Either you havent got the 5d3 focus set up right (it is very complex to start with) or you have a problem with the lens combination because the two focus systems are a country mile apart. With the 24-105 f4 on both cameras I find the 5d3 focus system extraordinary. 6d is fine but nothing special. I cant tell much difference between them noise wise. Both are excellent, although I find there to be more detail from the 5d3 despite the difference in pixel numbers being so small that it shouldn't be that noticeable.
I think you are probably right on the lens combo with the 50 1.4. My mark iii and 24-105 are also extraordinary. I did some more un scientific tests today with my 90 mm tamron f/2.8 (one of my least used lenses, but very sharp). The mark iii and that lens are awesome together. Also, after more noise tests I still think the 6d does a better job but there's not as much difference as I initially thought.