This depends heavily on the weather and where in Iceland you are. If you are staying in Reykjavík, you'll need to drive some distance out of town to truly get a good view of the northern lights. In more remote destinations it may simply be a matter of stepping outside. Of course, it could be cloudy the entire week. There are no guarantees.
Indeed, even with clear skies, the northern lights are fickle themselves and vary depending on how much solar activity there is. Your odds of seeing them improve as you go farther north.
Depends. If you intend to stick to the main highways and take care to monitor weather reports, a typical sedan will get you most places. We're pretty good about keeping the main roads open. If you want to go 'off the beaten track' you'll need a 4x4 and should travel with people who are familiar with the territory.
New Year's Eve, in Iceland, is typically one massive, uncoordinated fireworks display. You can even buy your own and join in.
All the usual attractions are also available, Blue Lagoon, national parks etc. There are also highland trips sold during winter (not right over Christmas) but weather may affect availability.
You will have no trouble finding accommodation. Summer is still the tourist high season and there should be plenty of rooms available.
Do note that Christmas in Iceland is a proper holiday. That means that almost every store and restaurant closes down in the afternoon of Christmas Eve (by 18:00 most everything is closed) and does not open up again until the morning of the 26th.
This does not apply to New Year's. Stores and Restaurants will be open throughout New Year's Eve and New Year's Day.
