Hi @Rowdythree -
We went to Reykjavik last December but at the beginning of the month. It was my second time visiting but the first time for others in our group. We all had a wonderful time.
We hired a small standard car- Mazda maybe? Wasn't a 4x4 but the tires were snow tires with studs in them to help with grip. I did the driving - normally my husband would drive in those conditions- and I was absolutely fine. The roads were all clear but it wasn't super snowy or icy while we were there... the car hire company do prohibit certain roads and as we drove past them we could see why. Those are the roads that are quite cross country but you don't use them if you're doing the Golden Circle.
We basically followed a route that the tours do.. having our own car meant we could stay as long or as little as we wanted- one area we literally jumped out of the car, took a photo and moved on again because it was raining so hard! Maybe take a thermos flask of hot drinks!!
When driving, just go at your own pace- people overtake you and just let them. We got overtaken by a coach and we were doing the speed limit! I guess local drivers know the roads and their cars.. but you don't need to rush and take unnecessary risks.
Food is very expensive- we took snacks with us in our suitcases from England so we wouldn't have to buy them whilst there.
I would recommend getting some of those clip on ice shoe things you can get from outdoors places here... we walked down to some waterfalls and it was very icy! They'll give you a little extra grip and confidence!
We found having the hire car cheaper for the airport transfer but check if your hotel has parking available!
Aside from driving, I wanted to let you know about a little gem we found by chance that wasn't advertised but we highly recommend, and I'm not sure we'd have got to without the car-
It's called the Röstin Restaurant- The Old Lighthouse Cafe and it's about a 20 minute (I think!) drive from Reykjavik along main roads (all clear!)
Went there for a birthday dinner. The food was basic but very tasty and the very friendly lady who runs it gave us the keys to the lighthouse and let us go over, climb up it and stand out on the platform next to the light.... it was brilliant and had the northern lights been active it would have been the most incredible spot to see them from! We didn't pay to go up the lighthouse but think it usually is an inexpensive paid for thing.... but I highly recommend it. It's right out on a limb into the sea (as you'd expect!) and so quiet!
I've added a screenshot for you of Google map details.
Enjoy your trip. It's a beautiful place with very warm and welcoming people.