Hey,
I went to Iceland in November a few years ago with my husband. We found most roads a few hours outside of Reykjavik to be closed and/or not safe for driving in small rental car (and we are from Canada so are use to driving in snow).
We were in the country for 7 days and based our whole trip from Reykjavik. We rented an apartment in the city from Apartment K and poked into a few hostels all of which looked great. During our week we spent one day doing a guided golden circle tour, another day we went horse back riding, spent a following day caving and snorkeling, and then a few days to explore the city (you can also go to the blue lagoon, do a northern lights tour etc.)...Basically, you can easily fill a week staying in Reykjavik and touring from there.
Also if you base your trip in Reykjavik there is not need to rent a car. From the airport to your accommodation you can take the Fly Bus (very easy to use, highly recommend), and anywhere of interest within 3 hours of the city you can find a guided tour. All the day tours we took picked us up from our accommodation and were good value for money especially given the weather/road conditions.
If you do want to rent a car and drive yourself I would suggest getting something a bit bigger than an economy car. Also we noticed most locals had spiked tires...I would request those on a rental car as well. Even in the city the roads were completely iced over.
Also a note on day light...when we were there our longest day was about 5.5 hours of "light" and it was not super sunny. The sun rose around 10 and set before dinner. It was an amazing experience to have such little light but, it would make driving difficult.
I hope you have an amazing trip! I thought Iceland in the winter was fantastic...no matter what you do on your trip Iceland will not disappoint :) Also I second the recommendation for warm cloths!
Sarah.