How important is shoe weight? Please do not confuse this with general weight on your bike or rotating weight. I'm onto about something completely different.

Logic tells me that everytime you do a pedal stroke you need to lift your foot along with most of the legs weight on the upstroke. The heavier your foot and shoe, the more energy you need to expend to lift it. On the other hand, a heavier foot and shoe will create more force on the downstroke.

Assuming you were on a home trainer with 10kg weights attached to each of your feet how would it effect your power output and pedalling style?

What if you ride out of the saddle? High cadence vs. low cadence?
You are correct that the weight of the shoes cancel out each other for the rotation part. For an extreme example of the same effect, check out Falkirk Wheel, which lifts 250 tons of water and boat using same amount of water as counterweight. 

The shoes are not usually centered on pedals, but the axle is under the ball of foot. If hypothetical heavy shoe has even weight distribution, you will have to support the weight of the heel. That weight does not do any work, and it would be possible to design a gear train similar to Falkirk Wheel that maintained the pedals level without outside forces.