FWIW there is no need whatsoever for a 4×4 on Route 1, the ring road. If you are hiring a vehicle you will not however be able to obtain any insurance if you take it onto any of the ‘F’ roads.
So, unless you are planning a crossing of the interior, eg via Kjolur or Sprengisandur I would bear the above in mind, it can make a big saving.
When all is said and done though I have seen many ‘normal’ cars (2wd) on the F35 (Kjolur). I would happily drive in a Golf or whatever to Hveravellir https://www.google.co.uk/maps/@64.8658825,-19.5525048,366m/data=!3m1!1e3. There is but a single simple ford to manage. The last mie or so, from the F35 main drag, to the hot springs is the most testing but take it slow and easy- no probs in my experience. But no insurance available, so a risk you need to weigh up.
Do not attempt to drive the Sprengisandur without a 4wd. I was escorted across a thigh deep ford by the Mountain Rescue guys once, right by the Nyiadalur Hut. Yet another time though it was barely ankle deep.
http://www.nat.is/travelguideeng/plofin_sprengisandur.htm
There are so many ‘iconic’ place to visit- but try to get away from the main tourist drag. My best experiences have always been in places seldom known. Try a cup of coffee in a quayside cafe in a tiny southern fishing port. http://www.tripadvisor.co.uk/Restaurant_Review-g608874-d2264342-Reviews-Bryggjan-Grindavik_Southern_Peninsula.html
or the best campsite in the world, very difficult to reach but well worth it. Pakgil. https://www.google.co.uk/maps/@63.5303455,-18.8987267,4127m/data=!3m1!1e3