What is the meaning of the Dirac equation obtained from the classical QED Lagrangian?

When I start from the QED Lagrangian density:

$\mathcal L_{QED} = \bar \psi(i\not D-m)-\frac{1}{4}F_{\mu\nu}F^{\mu \nu}\tag1$

by using the EL equations I obtain:

1.) The Dirac equation (I ommited the interaction term):

${\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m)\psi }=0\tag2$

2.) The Maxwell's equations (I ommited the interaction term):

$\partial_\mu F^{\mu \nu}= 0\tag3$

My question:

What kind of equation is eq. $(2)$? Classical or quantum?

It can describe quantum phenomena (at least it was introduced by Dirac historically to do so)...

But I obtained it from the principle of stationary action, so it must be classical field equation.

1.) What eq. do I get when I quantize this Dirac field theory than?

2.) Does the classical field Dirac equation have any physical meaning?