When fighter jets fire a missile, do they observe any kind of reaction force due to firing the missile? If yes, how is this observed by the pilot? If no, why not?
No, rocket missiles are recoil-less weapons.
When firing a gun, the charge burns inside the chamber and the generated gasses do not escape before the shell leaves the barrel, so the reaction force is transferred to the body of the gun.
When firing a rocket missile however there is open space behind he missile and the reaction force only accelerates the generated gasses that are ejected off the back and does not affect the launching platform significantly. This is true for missiles launched from tube (like Bazooka, TOW or Hydra 70), off a rail like the wing-tip sidewinders or from free fall like most of the heavier air-launched rockets.
The lack of recoil is an important advantage of rockets. Bazooka can be fired from shoulder while anti-tank gun with comparable explosive charge is heavy device and needs good support to handle the recoil.
On the other hand rockets need more propellant than comparable gun shells, because the generated gasses leaving at high speed carry away a lot of kinetic energy most of which would be available for the shell in a gun.
Ignore the actual purpose of the video and why the pilot is firing in the first place.
You can clearly see that the missile separates from the aircraft before igniting the thrusters. Since the missile starts its thrusters without contact to the aircraft, no recoil is experienced.
In some early jet fighters such as the F94 Starfire, air to air unguided rockets were carried in launch tubes that were built into the aircraft structure and closed at the rear end. Launching the rockets could produce enough recoil to cause serious control problems, and this type of armament fell out of favor as a result.