Could be that your motor has a rev limiter. Does it only do it in park or neutral? My grand prix has a rev limiter of 3000 rpms. It seems like the motor starts stuttering like you are hitting the gas an letting of repeatedly. This should only happen in park or neutral though. Could be worth taking a look at though.
mass air flow sensor
because the engine needs to have a 20 seconds RPMs
There must be a "check engine" light on. Start with investigating the DTC's (Diagnostic Trouble Codes).
Chances are it is a stuck fuel injector, or possibly a vacuum line leak.
Check the engine for proper cylinder compression. If the pressure is low then a blown head gasket could be leaking and causing the problem.
If it's cold outside, it's because ECU use ambient temperature sensor and coolant sensor to manage RPMs. After you start the car the coolant temperature is low and ECU sets RPMs very high, when it gets warm RPMs drop down until engine reaches operating temperature.
It will if you are free reving the engine with out it being in gear. If you are driving it hard down the road and it is kicking in then you may need to have the engine scanned with an OBDII engine scanner and make sure you don't have a problem in the system.
because the engine automatically revs up when it encounters the resistance of the ac clutch kicking in so it doesnt die or put un-needed stress on the engine at idle speed
check engine mounts
check for a vacuum leak on the throttle body
When you slow down in a vehicle without stepping on the brakes, the RPMs typically increase because the engine needs to work harder to maintain a consistent speed. This is due to the transmission downshifting to a lower gear to match the reduced speed of the vehicle, increasing the engine speed.
You can do it at pretty much any RPMs, so long as your timing is right. I can hit gears upshifting at 1300 or 1900 RPMs with no problem. What the ideal shift range is will depend on the engine. As a general rule of thumb, most people say 1400 - 1600 RPMs... personally, I like to shift a bit higher, and drop back down into the powerband when I go into the next gear, so that the engine isn't as prone to bogging down.