I'm pretty sure that a liquid solution of aqueous ethanol, when frozen sufficiently will simply yield ice and liquid ethanol; although this freezing point will be lower than 0 degrees (freezing point depression).
I don't believe that any of your queries pertain to chemical reactions (not any that I've heard of), air, alcohol, and water do not react significantly especially at lower temperatures...although there may be a slight pH dynamic.
Overall, iansmith hit the spot. Shaking expedites the cooling, the details are not so important...you're simply speeding up the energy distribution process. Heat (assuming a closed container) transfer is in the direction of lower temperature medium to the higher. The greater the distinction in temperatures, the faster the heat transfer. A temperature gradient as such that would exist if the martini/ice were left alone, will have a lower rate of heat transfer, enough so that the room temperature itself would have had quite a role in melting the ice.