*Sigh* I had a year tour courtesy of the Air Force in '79. I love that country. Sounds like much is still the same. Just a few observations that folks may be interested in but do research as this was almost 30 years ago.
There are tour companies based in Norway and Denmark that organize bus/bike tours of Iceland for a reasonable rate. They take you and your bike to a specific location and turn you loose. They provide maps and will meet you in a specified # of days at the next location. I believe you can cover Iceland in about 2 weeks this way.
If you want birds go to the north. If you want glaciers and rustic farm life go south.
A little known fabulous location is in the south at Skogafoss. There is a dirt path to the right of the falls that will take you up to the top. We actually took a 4 wheeler up there. From there you can follow a stunning canyon all the way back to the glacier. Do NOT take the 4WD across any of the streams up around the glacier - they are full of flint and will shred tires (you might well ask how we know this! ;-)
At the mouth of the canyon you can climb down and make your way all the way back following the glacial river. There is some stunning scenery, flora and fauna down inside as it traps the heat.
Anyway - wonderful country, wonderful people. Hitchhiking and backpacking didn't used to be any issue at all. One thing to remember is that many families maintain their family farms in the country and retreat there on the weekends. So you can easily hitch rides from the major cities to remote locations and get a free tour guide to boot. Just be sure you're available on Sunday to catch the traffic back in to the cities. Icelandic hospitality is almost embarrassing (for us) in how much they give and share. Most Icelandics speak 5 languages and English is virtually universal.
Y'all take care!