Is ... GPS data ... fused with the accelerometer data?
Yes, many aircraft use sensor-fusion techniques so both GPS data and accelerometer data effect the estimated X, Y, Z position.
Often they use a Kalman filter to do the data fusion.
( kalman-filter; Why do I need a Kalman filter? )
Measuring X,Y,Z accurately for each photo is important for assembling photos into an aerial photographic map. (a) (b)
Low-cost GPS receivers typically produce a X,Y,Z position once a second with an accuracy better than 20 meters. That is "good precision" for photographic mapping with a small RC aircraft or finding the destination runway in a manned aircraft. It's pretty much useless for a quadcopter flying in a room or a small field less than 20 meters on a side.
(Some GPS receivers are available that use DGPS and other techniques to get better accuracy, or generate a position many more times per second, or both).
How do ... quadcopters measure positions (X,Y and Z)?
They usually don't.
My understanding is that most quadcopters aren't doing aerial mapping or fully autonomous motion, so they don't bother calculating XYZ position. (c) (d)
As you already know from that video,
quadcopters use the accelerometers to improve their estimates of the aircraft orientation (its pitch, roll, yaw).
The quadcopter needs to know the orientation in order to stay right-side-up and to avoid spinning uncontrollably.
How do ... quadcopters measure positions (X,Y and Z)?
The people at the GRASP lab at U Penn measure XYZ position of their quadcopters using lots of expensive motion capture cameras bolted to the walls. They produce lots of really impressive-looking videos.
A few cutting-edge research projects are trying to measure XYZ positions using cameras attached to the quadcopters.
("sFly Quadrotors Navigate Outdoors All By Themselves")
( Track a moving object )
("Encouraging Developments In Quadcopters (w. Lots Of Video)")
Before GPS, commercial aircraft used airport radio beacons to do instrument landing navigate.
I see some people talking about related techniques for radio beacon localization for quadcopters.
Ian suggests tagging each landmark with a unique augmented reality glyph.
Some people find it more practical to use a camera on the quadcopter to detect and estimate relative bearing to a simple red rectangular landing pad.
("Autonomous Quadcopter Docking System")