Any question answered...

rent a banger from sadcars
if you get something else get the max insurance as the gravel will scratch it up

If you’re just staying in Reykjavik, with just two people I found it more cost effective to use Reykjavik Excursions to do the touristy stuff and not have to worry about parking in the city centre.
We went in April and I wouldn’t have wanted to drive in the snowstorms or late enough to catch the aurora borealis (or both).

Yes we hired a medium-sized Chevrolet petrol hatchback. You don't need a 4x4. No hire cars are permitted to go off-grid. It's a really good idea to obey this.

We went in March a few years ago. Hired a 4x4 Skoda estate. It was pretty hair raising to see other people in tiny hired hatchbacks swerving across icy roads in blizzards. In the towns and cities the roads were salted and cleared but further out not at all. We felt safe in the Skoda.

It was incredible to come and go from our hotel, deep snow could be drifted from one side of the road to the other in a couple of hours, so the landscape could keep changing.