1. Can the water quality from the well change over time?
2. Is it possible the iron we are now seeing in the water is actually rust from inside the existing pressure tank and not deposits from the water.
3. How can I flush/clean out the iron deposits in the pressure tank?
Take everything apart and clean it out.
Also the existing pressure tank is an air to water (bladderless) tank. After removing the hydro charger, the system has no air recovery system (I don't think well has bleeder valve).
1. How long will the air in the pressure tank last before the tank waterlogs?
Never as long as you think it will. And just one day of a waterlogged tank will kill the pump.
2. Can I get by with adding an air input into the system to recharge air into the system ("schraeder valve") and keep the air to water style tank.
Yes if you use a compressor with a float in the tank to work automatically. Otherwise you will forget, the tank will waterlog, the pump will burn up, and you will wish you had fixed it right the first time.
3. Should I replace the tank with a new bladder tank?
Not if you want to keep adding air to get rid of the iron. Although you could use a bladder tank and never have to worry about waterlogging, if you use something like the Sulfur Eliiminator to get rid of the iron in the well itself.