When is the best time of the year to cut and cultivate copper beech? Where is it best to make the cuts? Thank you!

uhm.. Your question is very unclear.

You can trim beech in winter. Just keep in mind it will not easily create buds so trim to a place where you still have buds oryou risk losing the limb alltogether.

I meant what month, in the midwest, is it best to make cuttings for propagation from a fully grown copper beech? Where is it best to make the cut: where the branch is still green? X number of leaves down?

as far as I know, beech does not really grow from cuttings.