Does the free will exist? No present research provides any evidence for or against. The idea that the free will does not exist is based on a lack of understanding how complex dynamical systems like the brain work.
Neurons are not associated one-on-one with a specific function. There is no such thing as a mirror neuron or the mirror system, just as there is no Jennifer Aniston neuron or neurons for a particular emotion or a particular movement. Just as a swallow in a flock is not a “flock swallow”, a neuron which is involved in seeing somebody else perform one of our actions is not a mirror neuron.