Honestly, I would only do one day in Reykjavik and the rest out of town to see more of the country. if you are flying in from the US/Canada on an overnight flight, you need that day to get over jetlag, so the day in Reykjavik is perfect for that. We have done the Citywalk "free" walking tour a few times on arrival day and it is a great introduction to the city/country. They book about a week in advance and they book out fast! So mark it on a calendar if this is of interest. The tour is free, but you tip what you want. Another great tour for arrival day is the food tour. Again we have done it a few times and it too has a lot of information about Iceland.
Both Blue and Sky are good, we have done both numerous times. Just do what fits your plans better. There are others as well such as in the Golden Circle and then just north of Reykjavik such as Hvammsvik (my all-time favorite) and Krauma. In the GC there is Laugarvatn, Secret Lagoon, the brand new Laugaras and then rustic ones. They are all great, it just depends on what fits your itinerary.
I would rent a car upon arrival and drive into Reykjavik. Try to get a hotel that has free parking such as Center Hotel Laugavegur or Center Hotel Midgardur. The the next morning head out. With 4 days, but not knowing what time your flight is, my general advice would be a nice up on Snaefellsness, a night in the Golden Circle and a night near Vik. Of course that all depends on flight details. You would not want to be in Vik the night before your flight, so maybe swap that with the GC lodging just to be safe in case of any road closures.
Having a rental car gives you so much freedom/flexability.