The Cessna you are flying in FS has what I believe is called "Positive Static Stability" or simply Positive Stability. This means that the aircraft is designed to precess to the orientation that it was previously at, after certain lateral and longitudinal movements.
The F-16 has Negative Stability, which I believe means that it basically points where you tell it to, and is not designed to be very forgiving. (Helps with manueverability I believe. Neither of these stability concepts directly involve computerized trimming of the aircraft.
You are correct about positive static stability. Though neutral static stability is where the aircraft "stays where you put it". And negative static stability means that it will diverge, requiring constant input to keep the airplane flying straight. The reason unstable aircraft usually have an FLCS is that the pilot would otherwise have to devote all his attention to keeping the aircraft pointed forward or not be able to react quick enough to fly the airplane at all.
To answer Question 1:
The FLCS on the F-16 does not automatically trim the airplane, but instead moves the entire control surface. Unlike a Cessna 172, the F-16 pilot has no direct connection to the control surfaces at all. Instead, the pilot is basically commanding a pitch or roll rate using the stick. The FLCS responds to that command by moving the control surfaces.
The difference can be demonstrated by flying the same simple maneuver in the 172 and the F-16.
First the 172:
1) Accelerate to a good cruise speed and trim for level flight.
2) Once stable, reduce the power to idle and try to maintain a constant altitude.
3) You'll find that as the aircraft decelerates you will need to pull back on the stick (adding more up elevator) to maintain level flight.
4) Eventually you will stall and no longer be able to maintain altitude with any amount of up elevator.
Now the same maneuver in the F-16 (I don't know how well Falcon 4 simulates the control surfaces. If you don't see this in the sim, trust me that the real aircraft does it)
1) Accelerate to a good cruise speed and pitch the aircraft for level flight.
2) Once stable, reduce the power to idle and don't touch the control stick.
3) As the aircraft decelerates the FLCS will command the tail surface to move elevator up, like the Cessna, to maintain level flight. But it does so with no input from the pilot.
4) As you approach stall the FLCS will reverse and start reducing the up elevator to drop the nose and prevent the stall. (Note: I don't know the particulars of the F-16, but it might have too much idle thrust to stall when clean at low altitudes. You can try dropping the gear or performing the maneuver at a higher altitude if you can't get the airspeed low enough.)