Autoland isn't a matter of simply pressing 'land'. The airport and aircraft have to be specially equipped, and the pilots properly certified. Very new, and older very large aircraft (Boeing/Airbus) are equipped with Autoland.
The aircraft has to support it, but the airport/runways no longer need to. ILS is being deprecated for synthetic glidescopes using GPS, as your precision is higher. DGPS is installed at the airport for increased position resolution, but the aircraft can fail back to WAAS in the event DGPS is unavailable.
They do. Autoland requires a CAT II/III ILS. WAAS is very accurate and will surely be used for this purpose in the future, but it isn't at the moment.