Android - How does the GPS work on Android phones?

GPS itself does not need a network connection to any server to work. It just needs a signal from a least 4 GPS satellites to calculate a good position fix.

There is also the GPS Almanac, which basically provides future data on where the satellites are expected to be in a given time in the future, usually up to 7 days. This can be used to speed up the position calculation. The Almanac is broadcasted by the GPS satellites and it takes up to 15 minutes to download the full Almanac via the GPS system. That's why there are also Servers on the Internet that provide the Almanac data in an more convenient, faster way. Maybe that's what you meant by "GPS Server". Android makes use of that too.

GPS - all GPS systems get signals from the satellites. They never send data to the GPS satellites. Your phone would need a much larger antenna to send a signal to them.

Systems like OnStar do send your position back to their servers. Google servers do get your position information so they can calculate directions and pull in the tiles for the map.

The US government does not charge for the use of GPS. It is global.

GPS is a passive system. There is a constellation of satellites in orbit, with atomic clocks on board, that essentially just shout out their identifications and the time, for anybody who will listen. A GPS receiver listens for the signals and uses the differences in times that it's hearing from the different satellites -- due to speed-of-light delays and relativistic effects (!) -- and knowledge of the orbits of the satellites (also broadcast by the satellites), to triangulate its position on the earth.

The signals from the satellites are just radio. There is no "network", phone connection, or any two-way communication necessary.