Civilian GPS receivers
First of all what was encrypted but now isn't is SA. What SA (selective availability) is, is the wobble of the satellite in orbit. Even though a satellite is in a fixed orbit it wobbles in space. Since your position is calculated using no less than three satellites (four if you need attitude) the compounded wobble of the satellites is what causes the error in position. The reason civilian receivers don't work at high speeds, besides there being laws against US companies selling them that work at high speeds, is the cost involved with processing power needed to calculate the signal changes at high speed. Also the antennas for high speeds are not like the ones that operate at low speeds. All receivers use the same information it was only the availability of the corrections information that has changed.
When it comes to differential GPS there are two methods. The one most people know about involves a base station that knows where it is located and can calculate the satellite wobble and transmit that information to the roving one. The second version involves a single GPS receiver with enough channels to handle processing the information from several antennas at the same time. Because it is a geometry calculation each antenna receives slightly different signal timing information from a satellite. By doing some matrix calculations on the data the wobble can be calculated real time for each satellite. Aircraft have an easier time doing these type of calculations because the satellite constellation is not obscured by items close to the horizon that block the signal. The better the satellite geometry (spacial positions) the better the calculations that can be performed. The differential GPS class I took about 10 years ago was taught by one of the system creators. In the test he and his students performed they attached several antennas to an airplane and then flew the airplane around before doing the post processing on the data collected (this was before the civilian receivers we have today which can process the data real time). they couldn't figure out what an error they were seeing in the data was until they realized that it was actually measuring the yaw of the wings where two antenna were located.
Spoofing is done using devices call pseudolites which are also located sometimes around airports to provide ground based GPS information for landing purposes since the satellite geometry, and precision, is worse the closer you get to the ground due to signal interfering items (the geometry of the tracked satellites determines the accuracy of your position). The newer surveying grade receivers are being made that not only use the GPS but the GLONASS constellation as well to provide better world wide geometry. For ground based location you only need 3 satellites in good position for attitude positioning you need at least 4 in good positions.