How do pilots maintain level flight after launching missile?
It's kind of a mix of the two. Flight controls on aircraft have what's known as trim. All (or nearly all) airplanes have elevator trim; many also have rudder trim and aileron trim. The trimmer for each control is used to balance constant forces needed on that control.

To take an example that's more widely applicable than launching missiles: when you increase power (to climb or to reach a faster airspeed), the airplane will have a tendency to pitch up, so you might find you need to push forward on the stick to maintain the right attitude. By moving the elevator trim forward, you eliminate the needed force, effectively changing the neutral position of the stick.

On early aircraft, the trim often worked by having springs connecting the control wires to a large lever. Working the lever pulls the spring in that direction, applying a force to the controls. On modern light aircraft, there is a trim tab on each control surface. The trim tab is like a tiny control surface on the control surface. Moving the trim tab to one side or the other provides a force on it, replacing the force the pilot would provide through the controls.

Fly-by-wire aircraft, such as military jets and large passenger airplanes, have electronic trim. Since the pilot doesn't have to put much force on the controls to work them anyway, trimming just resets the neutral position of the stick as you describe. Because it's electronic, there can be different ways to control it. Some aircraft still have small levers to set the trim; some have up/down buttons in reach of the stick (such as the Su-27); and yet others simply have a single button that trims to the current position of the controls (the A-10). As Mr Çetin points out, some very modern military jets take advantage of the possibility of computer control: the aircraft is aerodynamically unstable, but a flight control system manages keeping stable flight.

Returning to the particular case of the F-18, it doesn't re-trim itself. When heavy ordnance is launched, the aircraft will roll to one side as you suspect, and the pilot has to compensate with the stick and pedals. Once he has stablised the aircraft at the desired attitude and airspeed, he can adjust the trim using a hat switch on the stick.