I am no pro by any stretch of imagination. So I'm not really sure as to what's going on with your camera's Live View focusing. All I know is what I read up on the subject while dealing with my D800's focusing issue in Phase detect. All the articles I read mentioned that Live View focusing should be precise and if it is not, then there is something wrong with the camera.
Another thing to check out - are your lenses good and sharp? Do you need to do micro-adjustment for them? Maybe there is something in this area?
As for speed of actuations in Live View mode. I did notice that with my camera Live View shots are much, much slower. Not sure as to why.
Good luck with figuring out the problem. Hopefully there is nothing wrong with the camera.