A lot of questions! Let me try a few answers...
Absolutely I'd say stop in Iceland. Every place in the world is unique, but Iceland is more so, geographically, in climate, and in history. Half a week (or even a week) won't do more than scratch the surface, but you'll be able to visit incredible waterfalls, climb on glaciers, see evidence of recent volcanic activity, and realize that under it all is a huge pool of thermally heated water that provides over 70% of the nation's energy.
If that weren't unique enough, Iceland was almost complete isolated for over a thousand years, starting with a Scandinavian culture that then developed isolated from the rest; the differences are as fascinating as the similarities. In Rekyjavik, where most of the population lives, there are several great museums illustrating this history...including an excavated 1000-year-old long hall just under new downtown buildings. The isolation has also left Iceland with unique breeds of horses and sheep.
With a half week, you'd probably do well to take advantage of some of the bus tours offered by several operators; they pick you up at your hotel, and return you at the end of the trip. Don't know your age...but Reykjavik seems to be a town for the youngish party crowd, especially on weekends. Us older folk skipped that, but found quite a few good and interesting restaurants.
There are plenty of hotels in Reykjavik at different price ranges (including a new one just opened near the bus station that combines a luxury hotel with an almost hostel-like one in the same building). We were able to rent an apartment right on Laugavegur, a main shopping street, for less than most hotel rooms.
I'm kind of running on here, so write back if you want specific tours, or restaurants or apartment details!