If you are not used to driving in snow, or ice, or in windy weather, and are unsure of your own capabilities I wouldn’t recommend driving in Iceland during winter - at least not if this is your first visit to Iceland and you haven’t got an idea of what the winter weather can be like.
If you think you are up for it, then just make sure you drive slowly and carefully, and I’d recommend renting a 4x4 car in wintertime which will make you feel more at ease.
If you are coming in September, October or April, then there will most likely not be any snow or ice on the roads, so you should be fine. November until March is colder and more snow heavy generally.
Driving the Golden Circle, Snæfellsnes peninsula and the south coast to the glacier lagoon are the easiest routes to drive in the wintertime as they are in the south of the country and the roads are the least likely to be closed due to bad weather. Perhaps you should go on a tour of the south coast and Snæfellsnes, and then maybe rent a car to drive the Golden Circle yourself.
All rental cars have winter tyres during wintertime. All cars also come with Collision Damage Waiver insurance, which lowers your self risk in the case of an accident. Additional insurances are available to add to your booking, including one that lowers even more what your self risk is. If you get stuck you should call your car rental and inform them about what has happened and see what they have to say, each incident varies. If you crash your car or there is an emergency then you should call Iceland’s emergency number 112 immediately.
As long as your driver’s license is European, in English or translated into English then you should be fine. You have to have at least 1 year experience of driving and be at least 20 years old to rent a car in Iceland, or 23 years old to rent a 4x4.
Hope that answers everything!
p.s. You can read this ultimate guide to driving in Iceland for more information!