No problem Felix, but I'm just a simple controller - I can give you my understanding of how the aircraft is operated but I stand to be corrected by pilots who do that bit for real! The cat III bit is defined by the decision height for the approach (i.e. the height at which the crew must decide whether they can see enough to complete the landing safely and successfully) - below 50 ft, by definition, is cat III. Whether the flight is able to make an approach to cat III limits is determined by many factors including those I mentioned earlier. For a cat III approach I believe all landings will be completed automatically (including keeping the aircraft straight during the roll-out). For a cat II approach I believe that different airlines have different procedures depending on the aircraft capability and equipment certification etc. - as I understand it, some aircraft will be landed automatically whilst others will be flown manually by the pilot for the final portion of the landing from some point at or beyond the decision height. Again there will be many factors that affect what will be permitted and what approaches are available, for example, it may be that the signal indicating the centre of the runway will not be accurate enough for roll-out guidance which, if I recall correctly, means that only cat II can be approved (I would need to dig the books out to be certain so please don't quote me on this one!).
As for what goes into the FMC - I'm sure a pilot will correct me if I'm way off the mark here but for the sort of operation that I think you have in mind the aircraft will be set up for an instrument approach and will only make a visual approach if the conditions are suitable. The fixes used for an instrument approach will still be useful as a guide to the pilot but a visual approach is essentially navigated (officially anyway) by looking out of the window so there is no need for any new waypoints to be entered (and just after going visual is unlikely to be a good time to start entering fixes!).
The ATC side is quite straightforward. In the same way that the aircraft is set up for an instrument approach until a visual approach can be made, ATC will assume that an instrument approach will be flown until the pilot requests a visual approach. I won't go into the details here but ATC may still have to separate an aircraft making a visual approach from other aircraft and if there is no way to achieve this an aircraft may be refused permission to make a visual approach by ATC.