I don't know offhand, but when I get a chance, I'll check the brochure stand at the ANC airport to see if they list anything for tour operators in Fairbanks (I work at the ANC airport).

This is most assuredly the off season, so car rental rates *should* be pretty dang cheap (well under $30 including taxes). Fairbanks is pretty spread out, so I would definitely look into renting a car--unless you're under 21 (which I take it might be possible, since your username is, after all, premierstudent). Still, there are small mom-and-pop rental agencies that might rent to people under 21, though (search the acsyellowpages.com online phone book for Fairbanks).

Although the majority of the tourist industry is shut down for the winter, if you're going to be in Fairbanks for more than just overnight, take a look at the tours offered at UAF (here, here and here). Some special tours can be arranged during the school year with advanced notice. I strongly recommend checking out the Geophysical Institute (including the Alaska Volcano Observatory and the Alaska Earthquake Information Center) if you are at all interested in science. The University of Alaska Museum of the North is fascinating if you're in to anthropology or the arctic in general--it's the best museum in the state.

As far as your day trip to Barrow--I don't know too much about the area, but Barrow's taxi drivers are legendary (there was a big Anchorage Daily News special on them last year). Getting around town via taxi shouldn't be too much of a problem. I'm not sure how far Point Barrow is from the town (the town itself is pretty small), but word has it the taxis are pretty economical.

Also, for aurora viewing opportunities, look at the link I posted earlier to the UAF Geophysical Institute's aurora predictions. On the right hand side of the page is a long-term forecast. This might help you pick some dates on which you're likely to see some good aurora. (The prediction isn't meant to be super-accurate, but it is based on known hot-spot activity on the sun and is calculated using the sun's 28-day rotational period and serves as a general guide for trip-planning purposes.)