If you can forgive the self-promotion, I wrote a "things I wish I'd known" post about visiting Iceland which might be of interest to anyone planning a trip there: https://blackshaw.substack.com/p/iceland
A correction to "Unless you have a 4x4 it’s illegal to drive on certain mountain roads":
Legally you can drive on whatever road you'd like in whatever vehicle you'd like, but coverage in case something goes wrong is probably not included in your rental company's coverage. The "4x4 only" signs you referred to are meant to discourage tourists who don't know any better.
Depending on the road, weather, your driving skills etc. you'll easily be able to traverse certain "4x4 only" roads even in a VW Golf. The trick is knowing which roads those are, and accounting for the tiny stream you passed on the way in potentially having turned into a river due to rains or snowmelt since you drove in.
> We booked a hostel in Reykjavik for the whole trip. This was, without a doubt, the biggest mistake we made. It would have been a much better idea to drive around the whole island staying in a different place each night. (Iceland’s main highway goes around the edge of the country in a loop.)
This is what my wife and I did a couple of years ago and I highly recommend it. It took us 5 days to circumnavigate the country.
If you can forgive the self-promotion, I wrote a "things I wish I'd known" post about visiting Iceland which might be of interest to anyone planning a trip there: https://blackshaw.substack.com/p/iceland