We drove! there a few years ago, in a Mondeo estate, from Carmarthen. Day 1, Home to Hadrian's Wall. Day 2, Hadrian's Wall touristy stuff, to Aberdeen. Day 3 Ferry to Lerwick, Shetland. Day 4-6 Shetland, including Mousa Broch- incredible! Day 6-8 Ferry to Seydisfjordur, via Thorshafn, Faeroes, DK. Day 8, Landed Seydisfjordur, Iceland. South Coast route to Hofn. Day 9- 12 Skaftafell National park. Day 13, collect brother from Kirkujarbaerklosters! near Vik. We now have 3 adults and two kids in the car, plus a roof box, four mountainbikes and a tag along! Car is felling OK though. Day 14- 17, visiting friends in Reykjavik, plus day trips to Golden Circle destinations- Geysir, Pingvellir etc. Day 18- Drive north to Blonduos, then a two day cycle into to interior to Hveravellir hot springs and then onto Hvitarnes.
Later on we drove to Myvatn, via Akureyri, went whale watching (twice!) from Husavik and somehow got brother John back onto a bus to Reykjavik and then to fly home. In the meantime I almost lost it driving a tad fast on the dirt road when I hit a sheep one night.
Then on to Jokulsargljufur national park for a few days- The Northern lights are amazing!
Two day drive from there via the northernmost point (not quite in the Arctic Circle) to Seydisfjordur, to visit possibly the best bar in the country If only I could remember its name. Just some bloke's converted house. We went in, got beer, sat down, it was magical.
Then last days, on the ferries via Faeroes to Shetland and the Aberdeen. Landed at 8:00 hrs, lunch in Parbold, home for tea. The sleeeeeeeeeep.
Damage to car after a month of silliness was new front bumper; new front suspension bushes a new set of tyres! And in all reality it really needed a respray. However, we sold it when it was raining- result
Have also crossed the island by bike and 4X4 a few times.
BTW, the F35 (Kjolur) is now possible in a car. Care needs to be taken in a couple of places but you should be fine. No major fords or nasty surfaces to be dealt with. Rock and Roll!