I spend 2 weeks touring around Iceland on my bicycle in 2004. Besides that, I have been to Iceland more than 15 times! – but no personal touring content yet… any questions? just ask! :-)
Take a look at the website of The Icelandic Mountain Bike Club, Íslenski Fjallahjólaklúbburinn. Their site includes a “Guide to cycling in Iceland” in english with a lot of valuable information about bicycletouring in Iceland.
If you are in need for a local bicycle store in Reykjavik, I can recommend Örninn
If you need to rent a bicycle while you are in Iceland, you can check this page out. https://www.bikerental.is/
Also take a look at the tour operator Reykjavik Bike Tours site. They offer guided tours and bicycle rental as well as Segway tours and walking tours.