Yes, rotation of the earth affects flight times.
Look at the flight times quoted for a trip from LA to London,
Outbound: c.10 hrs 20 mins
Return: c.11 hrs 20 mins
The distance is the same but the travel time is about an hour shorter. Aircraft cruise at very similar speeds (i.e. they don't fly any faster to get to the US) the world is rotating London to LA direction and hence it is rotating under the aircraft bringing the destination nearer to get to the UK while moving it away on the trip to the US.
With regards to GPS it's not specifically the time that the satellite has (i.e. 12 noon) but the accuracy that it maintains it, +/- 1 second every so many thousands of years. GPS relies on speed distance time calcs to give you the position relative to ground stations that keep the GPS satellites updated with their own position relative to the globe.
I hope that is as clear as mud.
The tomorrows world experiement was to see whether time slowed down the faster you travelled, so if you travelled at the speed of light it was hypothesied that time would stand still. I don't know how far that research has gone though.