How do fighter jets stop pitching motion after a elevator deflection? I'm working on a flight simulator , In which the jet has only ailerons , rudder , and two elvators at the back . When I nose up my aircraft via elevator deflection , it just keeps pitching up ( even when I let go of elevators and let them back to neutral)  , after going pitch max , it starts pitching in opposite direction and then back and forth . These oscillations take lot of time to die out and the motion  of my jet is like a sine wave going forward .
I'm using real forces of lift and weight on the aircraft body.
So my queries are ,

How does a fighter jet locks it's nose at a particular pitch angle ( the moment the pilot let go of the deflection) and  does it so smoothly and slowly , that it's possible to aim at whatever direction you like and hold it there , even when flying at 600KM/Hr.
Because when I let go of my elevators , entire plane just goes into oscillations and it becomes tedius task to focus on killing those oscillations .
A cheat that I'm thinking of is , let's say pilot makes +30 degree deflection to elevators and let go , then the computer will immediately put the elevators to -30 degreee deflection . This will exactly cancel out the rotational momentum built up by the initial deflection .  But in real aircraft , it will take time to put the control surface in opposite angle . So how do they do it so quickly .
This person claims to be using a PID controller for pitch damping . How does a PID controller acount for the inherent delay in the actuators of control surface . Let's say controller decides an +10 degree is needed this moment to keep the aircraft stable at the current pitch , but the hydraulics will take their own time to put the control surface at that angle , by that time aircraft will have moved even more from the required stable position .

I know I'm asking too much in a single question . But I've been stuck on this problem for so long that only more and more problems have accumulated in this time .
This depends on what level of realism you want.
If you want high, save yourself a lot of work by using an existing dynamics model like the JSBSim. There are model templates for it that have all the needed functions for reasonably realistic model are predefined and you just tweak the coefficients to get the performance you want.
If you want low, just make the surface deflection correspond to pitch/roll rate rather than acceleration, because that's what actually happens to a level of approximation that is perfectly fine for an arcade gamer.
What really happens is that when you deflect the control surface, the rotation will change the angle at which air hits the surface, and that will negate the effect of the control surface, so the aircraft settles at fixed rate of pitch/roll corresponding to the commanded displacement. And when the pilot centres the surface again, the aerodynamic force still exists that countered the deflection, which will now stop the rotation again.
This is fast enough that pilots normally don't need to stop the rotation with counter-movement of the controls except sometimes for fast rolls in aerobatics.
Regarding PID controller, a “position-integral-derivative” controller, the point is that the derivative term predicts the response delay, while the integral term compensates for any bias. The disadvantage is that if you tune it to have fast response, it always overshoots the target value, because the integral term will overestimate the bias. But it is simple to implement and does not require much modelling, just tweak the numbers until it works. The JSBSim provides PID controller (and some other simpler controllers) for the model definition.