I have been flying a Cessna 172SP with an old-school Bendix stack and unlike the Garmin GPS and S-Tec autopilots that I am used to flying, I need to enter the altimeter setting into the KLN 94 GPS and the KAP 140 Autopilot. Most of the GPS systems in GA aircraft use Barometric Aiding to assist in detecting integrity anomalies. Most Garmin units interface directly with the altitude encoder to provide baro-aiding, but other units require the pilot to enter the local altimeter setting. AOPA has an article that lists non-WAAS GPSs that have baro-aiding. WAAS-enabled GPS units do not rely on RAIM so they do not rely on baro-aiding.
As usual, John D Collins has an excellent discussion of this.
With WAAS, the calculations are determined by the WAAS system as each ground station that measures its position via the GPS satellites is at a know location, and the error can be determined one satellite at a time. This integrity data is uplinked to the WAAS satellites and is broadcast as part of the WAAS correction messages. So the WAAS system itself is determining the integrity information via explicit measurement and passing it on to the WAAS receiver to determine the HPL.
The autopilot has an altitude capture and hold function where you can select an altitude and climb or descend rate and the autopilot will adjust the elevators to capture the rate and altitude. You still need to set the appropriate power for a climb or descent. You need to enter the altimeter setting so it knows where to stop or hold.