If the light intermingles with the mirror, then how much energy is lost in the reflection? Also, if E=h/lambda and h is constant, then the speed of light must change.

There's definitely always a finite amount of energy lost in the mirror due to the intermingling. The best mirrors, which I believe are dielectrics tuned to very, very specific wavelengths, are around 99% efficient. A typical bathroom mirror is much more lossy than that.

And yes, the speed of light does change within a medium other than free space. The change in speed is what's responsible for refraction.

Again.
Thanks

Do you mean the reflected wave is emitted by the quantum junps of the electrons changing energy levels, in the atom?

I'm not really sure. It may ultimately come down to that, but I think the process is significantly more complicated, involving photon-phonon interaction and all that good stuff. I'm sure if you Google it, you can find way more info than I would know.