6 days based in Reykjavik

We just grabbed an Icelandair special, and will find ourselves in Reykjavik for 6 days in March. This trip includs a Golden Circle Tour, but nothing else. Any and all suggestions for what to do for the other 5 days would be great. Should we rent a car? Can we get around by public transportaion? Anything we shouldn't miss? I know that seeing the Northern lights isn't guaranteed, but if it's clear, where is the best place for us to view them? Any suggestions for dogsledding?
Thanks in advance for all your help and knowledge!