It is not only the heavy snow you have to worry about, but the winds can also be fierce. A strong gust of wind can blow a car off the road, and there are certain roads that Icelanders try to avoid when it is really windy out. Just listen to the advice of the locals and you'll be fine.
If you do get a car, make sure it is a 4WD manual car. It is the best thing for the wintertime here in Iceland. People driving 2WD or automatic in the city sometimes get in troubles due to snow. So if you are planning on leaving the city, get a 4WD Jeep/SUV manual.
I think you should be safe in the south of Iceland and also the west.
There are some beautiful attractions there. You can check out the winter self-drive tours here. https://guidetoiceland.is/book-trips-holiday/self-drive-tour-packages/winter-self-drive-tours
You can travel the whole south coast to Jökulsárlón glacier lagoon. On the way, there are some stunning sites such as the waterfalls Seljalandsfoss and Skógafoss. I just read a blog where a woman was wondering whether Skógafoss was more beautiful in the summer or the winter. I am sure this is the same for so many attractions in Iceland. Here is the blog https://guidetoiceland.is/connect-with-locals/jorunnsg/different-display-of-skogafoss-and-the-legnend-of-it
The cool thing about Icelandic winters is that they are really dark. The sun only comes up for a few hours each day. My favourite thing is to get a remote cabin somewhere and chill in the hot tub surrounded by darkness.
The nights can be a bit spooky. There is a horror film festival in Akranes, in East Iceland held in late October or early November. They have screenings in an abandoned factory, how creepy is that?!?! If you are a horror fan, it is perfect on long dark Icelandic nights. Check it out here http://frostbiter.is/
So, if you do get a car, make sure you listen to every advice you are given. It breaks our hearts every time we hear of a car accident (or death) that could have been avoided because people weren't careful. They maybe went out driving in conditions they couldn't handle, in conditions in which Icelanders know that nobody should go out.