You should note that in winter time is better a 4x4 car because the snow and in summer time is also better if you want to discover the hihglands where in my personal opinion is the wild nature of this country.
Me and my husband travelled to Iceland for our honeymoon in end of june 2013 for 8 days. We rented a car with the company you were talking about... www.reykjavikcars.com since they were the cheapest we could find. We didn't have high expectation but we got a new car with free GPS (very important in Iceland since not all roads are well marked!), something we did not expect but saved our trip!
Just few tips of advice, be careful not to cross rivers if you dont know what you are doing and ask locals to guide you where and how is the best way to cross. Also, get a 4 wheel drive, some of the gravel roads are tough on small cars. We had a Dacia Duster and it got us everywhere we wanted to go (plus the diesel was relatively cheap). But we did come across some tourists that had problems with their small rental cars, especially in Thorsmork. Last but not least, stay on road and prepare yourself for all kinds of weather!!:)
Hey! I was in Greece this summer. It was awesome, I went Greek Island Hopping found an e-pass with Hostelbay.com . I am thinking of going to Iceland this year. When did you visit? I am looking to spend around 6 days. I hope the weather is not too cold. 4x4 Seems like a good idea, thanks!