1) Technicaly, there are 2 possible locations that the same distance from each of 3 satilites, but the other point is far off in space. The 4th is vary useful for increased accruacy when you have a shitty ground clock, but it's not 'needed' it's it's loss simply results in lower accracy which can be compensated for by more readings over time. (Which can be vary useful when want accuracy and have a ground station that only get's to see a small chunk of the sky.)
2) They all have polar orbits that reach the same location 4 minutes earlyer each day. However, there horizontal ground speed relative to a stationary observer changes over the course of their orbit. (It's zero at the pole and maxes out at the equator.) And is also diffrent form the other satilites you are comparing it with.
PS: Yea, that picture was fairly bad, I was looking for an easy to read article not a picture. The classic picture of a constilation with fixed polar obits is vary misleading once you take the earth's rotation into consideration. It's rare to find something that shows both the orbit's and their horisontal velocity fairly accuratly without making it look like they interweave. I once saw an animation showing the orbit of 4 satilites and the area they could cover area just those 4 satilites could give you coverage of but it's far harder with just a picture.
Suppose you have 3 points in 3d space.
P1 = (1,0,0),P2 = (0,1,0), P3 = (0,0,1)
You know point the distance from U to P1 is x, from U to P2 is x +1 and from U to P2 is x + 2)
You know know U is on a 1d curve (c1) though 3d space. (Or as you say you don't know time aka distance.)
Now suppose P1, P2, P3 velocity is more than 2x that of point U and you have you have new offsets from U to {P1,P2,P3} you are now going to have a new curve (c2) in 3d space, but you know U's vector must take it from c1 to c2.
Now add another time cycle, and you know it's vector must take it from C1 to C2 to C3. As you keep adding C's you constrain both the valid vectors(direction and velocity) as well as positions.
Now measurement accuracy as well as change in velocity limit the accuracy but GPS satellites are moving a lot faster than you are so you can get reasonably accurate information this way. Edit: Also you can basically ignore the sections of the curve that whose altitude is unreasonable so the solution space is fairly constrained to start with and for each subsequent curve. But you can also use an internal clock/oscillator, accelerometer, gyroscope to further constrain reasonable accelerations etc.
1) The intersection of three spheres results in two possible points. One of which is most likely somewhere in space or deep below ground. So 3 satellites are actually enough in theory. Altough you get a [much] higher precision with more satellites.
I agree on the other points. They all have an inclination of 55° in regard to the equator. See Wikipedia [1] for details and a more serious illustration
You would need three if you had an atomic clock: if you knew the precise time. But in practice you do not. You have a good clock that has an unknown shift with respect to the clocks in the satellites.
So you can measure differences in times, but not absolute times. I.e. you do not know the distance from the gps to the satellites, you know the difference of these distances. So you need a fourth satellite.
This was designed because it is easy to make accurate relative time measurements, but hard to have an absolute time reference.
I know for a fact that some aircraft are able to operate on three by using altimeter information to resolve the ambiguity between the two possible points. It's not as accurate as using four satellites, but it's still pretty damn accurate. In a boat you could get similar results by assuming that you are at MSL.
When you solve 3 simultaneous equations in 4 unknowns there are far more than 2 possible solutions. However, you are correct that if you have a priori information about one of the unknowns (usually the altitude, since it's easier than carrying your own atomic clock) you can solve for the remaining 3.
You can get around that by starting with a few reasonable amputations. aka speed less than 2,000 MPH Acceleration under 3g within 40miles of surface etc. And then refining your estimated local time based on the reasonable solutions and then repeating this a few times with several data points to get a better estimate for location, velocity, clock drift, and time. This tends to works quickly because your location and velocity are fairly constrained and the satellites are moving very quickly. As for a ground station, the velocity is vary constrained and which makes all of this a lot faster and more accurate.
Don’t forget we have a lot more CPU to throw at these problems than they did in 1980.
2) They all orbit in the same direction.
3) There are a number of different orbits, but they are all inclined at the same angle.
The illustration on that page was, I'm afraid, drawn by someone with no clue what the constellation actually looks like.