Like you, when I first got the G2, I was a little disappointed in the demands upon the user of the auto-focus system. I think you have to accept that this is a semi-professional camera not really a point and shoot. Its focus system is very accurate, when it does pick up a focus, as you will see when you get some results back. The downside is that it requires an object in the focus window on which it can clearly pick up that accurate focus, particularly if it is over about 3-4 meters away, when the infra-red system is beyond its limit. The focus will just not pick up on horizontal lines, repeating patterns such as leaves or brickwork and very shiny objects. It obviously uses a similar algorithm to my RX as its digital focus assist also has these limitations. The camera will not fire the shutter on SAF setting unless it picks up a focus. On CAF setting, if it does not pick up a focus, it will fire with the lens focused at infinity. This is useful for taking into the sun (another circumstance it finds difficult) for landscapes.
You very quickly get the hang of focusing off centre, half pressing the shutter release, re-composing and then shooting. I prefer not to have the exposure locked by the shutter button but only by the AEL switch. That is, however, a matter for personal preference and is selectable on the custom functions.
The difficult one is when you are taking a photo in portrait with the camera turned through 90 degrees and there are no horizontal lines to pick up. You try turning the camera back to the normal position to pick up a focus on what are now vertical lines, half press the shutter release, then on turning back, you accidentally fire the shutter at about 45 degrees. You need, in those circumstances, to use the focus lock button on the back, which as I have rather arthritic fingers, find a little difficult but I am getting better at it.
You will find, on the assumption that all is well with your G2, that you get used to its idiosyncrasies after a few rolls. However, I would take it to your nearest Contax dealer and get them to have a quick look at it, which they should do on the spot. Hope all this helps. Wilson