If I leave my fan switch on "on" instead of "auto" and the temperature in the house reaches the desired temperature will the fan stay on and continue to push cool air out even though the thermostat turns the compressor off.
Yes, that's what "on" means. It means the air handler will keep operating, even though nothing is heating or cooling the air.
Whether this is a good idea depends on how your system is setup and what is going on.
If you are getting thermal separation ceiling to floor or upstairs to downstairs or sunny to shady side, it can be helpful at evening things out.
One person on here a few weeks ago had a problem where one side of their house was much hotter. So they had their air conditioning re-"balanced". Now the side with the solar load got a lot more A/C; this also meant the other areas got a lot less. With so much less air conditioned (dry) air, condensation started to occur. "On" might have helped them with their problem.
Yes. the fan will stay on until you switch it to a different setting. Even with the compressor "off" the fan will continue to run.