Below are the answers to your questions, respectively:
1) Reykjavik is a pretty nice and relaxing city. How much time you want to spend there, depends on what you like and what you want to do there. If you like going to the pubs and just relax for sometime, then I would recommend to stay for a couple of days in Reykjavik.
2) If you want to take the Route 1 (Ring Road), 3 or 4 days would be good enough to drive around and look at some attractions like (Snaefellness - mentioned in 'Journey to the center of Earth' novel/movie, Westfjords, Akureyri - Northern Iceland, Lake Myvatn, Vatnajoekull, Skogarfoss, etc). You can also do it in shorter time. If it was me, I would take more time and explore all the places, stay overnight at a couple of places where I can relax and visit some attractions around, etc. Apart from this Route 1 (Ring Road), you can drive around the Golden circle, which has some of the main attractions of Iceland (Geysir, Pingvellir, Gulfoss waterfall, Central Iceland, etc). Central Iceland is really exotic and it feels like you are on a different planet or moon. Some of the US astronauts (including Neil Armstrong) are said to have been in this part of Iceland, as part of a training programme, before the Moon expedition.
3 & 4) With a normal car, you can reach many important locations, but not all. Driving in Iceland is relatively easy, apart from a few gravel roads and some parts of Central Iceland. Anyways, you can still get to do these trips, as there are many offers (not that cheap), where they use those huge All-Terrain vehicles.
I've been to many parts of Iceland and have seen much, but not everything...As for me, the best parts of Iceland are Westfjords, Exotic Central Iceland, Snaefellness & surroundings, Akureyri, Lake Myvatn, Vatnajokull, Vestmanneyjar (Westmann Icelands - Volcanic Island on the South of Iceland), Grimsey (Island on the North of Iceland), Rejkyavik.
EDIT: If you would like to take your car to Iceland, check Smyril Line.