Go in summer if you want to ride, though. Ask me how I know that October will freeze your fingers off
I'd 100% recommend visiting Iceland in October in general. It's a lot less busy around attractions and the sea climate makes it not that cold (the main challenge usually is breaking the wind). Walking/driving around is absolutely no problem, and outdoors swimming is also fine because the pools are heated geothermally. Heck, glacier hiking wasn't a cold experience at all with fairly regular winter clothing (keep an optional rain layer in your backpack for the unpredictable weather though). There has been only one activity where the icey wind got the better of me despite adequate clothing (I don't know what more I could have put on, the only solution I see is electric heating or walking parts of the way to warm up).
If you go, take at least a full week excluding travel days and 'cheap campervans' was by far the best deal that I found for a vehicle (seemed almost too good to be true but everything was as advertised and the people super friendly).
The experience taught me a thing or two about weather at home as well. I don't like raincoat material so I never had rainproof clothes. Now it was kind of essential to prepare for that and I found there are more plasticy pants and jackets as well that are no problem. Layering and wind breaking is how one stays warm beyond staying dry. In the future I'm going to be enjoying the outdoors more during winters at home :)
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.
