Yes, I know you can do that, I even mentioned it :) But the person I was repying to claimed "3 satellites are sufficient to find your basic location and elevation" (emphasis mine) - i.e. they were claiming you could find the distance from the centre of the Earth, without assuming it.
No. You are finding your distance from the satellites. Not from the center of the Earth.
Umm... if you're claiming you can find your elevation, then you can find your distance from the centre of the Earth. It's more or less the same thing :-). I agree you can use the measurements from the satellites though (i.e. you don't need to know any information about your elevation a priori).
But since the 3-D positions of the satellites are known to a fine degree of accuracy, that's all you need.
I agree that the position of each satellite is known, to high accuracy. That's not the issue - the issue is that, if you have a signal from one satellite, that doesn't tell you your distance from that satellite. The signal basically says "I am satellite 7, I am at (position), and the time by my clock is (time)", and that is all your receiver receives. Receiving that signal tells you nothing about your position (well, possibly you can tell which hemisphere you're in from the fact that the satellite is visible at all, but that doesn't narrow things down much!). You only start getting information about your position when you have signals from more than one satellite, because then you can compare the timestamps - the difference in the timestamps gives you information about the difference in the distance from you to the two satellites.
And yes, 3 points and 3 distances is all you need to find a point in 3D space. It actually defines 2 points, but as mentioned before one is out in space so it obviously doesn't count. The other one is on the surface of the Earth (or in a plane, or whatever).
I agree that 3 points and 3 distances is all you need. The trouble is that the signal from 3 satellites only gives you 2 distances (unless, as jfengel mentions, you happen to be carrying round an atomic clock with you - which is generally not practical!).
You still need to think about probability, because the elevation model only tells you where the ground is - not where you are in relation to it. That said, the asumption "you're probably very close to the surface" is often very good, at least for most people :-)
No. Again, the orbits (and therefore positions) of the GPS satellites are known very precisely. As long as you can receive the signals from them, you could be above the ground, or below the ground, or even at the center of the Earth (at least theoretically), and still get your position and "elevation", even though it may be negative. Probability is not a factor at all, though accuracy certainly is. There are errors that have to be accounted for.
I mention probability because, even in the absence of 3 distances to give you a full 3D fix, you can use the fact that, statistically speaking, most people spend most of their time very close to the Earth's surface, to give a good estimate for one of the distances (i.e. estimate "I'm probably at or near ground level"), which then generally gives a pretty good estimate of your horizontal position. This is why even a 2D GPS fix is very useful in practice. But even if you have a signal from 4+ satellites, probability still comes into play: you have to account for radio interference, atmospheric effects and multipath effects, inaccuracies in the receiver's clocks, rounding errors, etc. etc., each of which adds uncertainty to the measurements and calculations and hence to the ultimate position readout.