you understand you have all sorts of clearance issues, weight issues, and upper room when the door is up. If you have a GDopener, then you also have that to deal with.
Now, the steel door flex's as you know, so any wood glued to the door might not stick over the long run.
Therefore, I would consider installing furring strips and fastening them to the door with small carriage bolts and washers and nuts, then attach the wood to the furring strips. This will allow the door to flex, contort and actually be independant in motion to an extent from the wood skin.
Your concern is to get a thin furring stip, but thick enough to hold the wood.
Good luck on this and keep us posted as to your plans and its outcome.