> 1) If you're standing at the South Pole, what time zone are you in?

The North and South Poles are of course at the meeting point of all time zones, so there is no official time zone for either pole. It tends not to matter much because there isn't a huge workforce in either place.
The Amundsen-Scott South Pole Station sets its working clock to New Zealand time. Other countries who have stations there use other time zones depending on their preferences.
But everyone in Antarctica uses GMT for scientific records and similar purposes.

Brunnen-"unless you're talking about the South Magnetic Pole, in which case your watch wouldn't be working anyway, because you'd be underwater"G