When we went we stayed in Youth Hostels the people there were very helpful, we planned where we were staying in advance online many emails between them & us, paid them, & collected vouchers at the airport which we exchanged at the hostels.
Think we stayed in Vik, Hvoll, Höfn, Eyrarbakki, Vestmannaeyjar & Reykjavík. All we very welcoming & amazing places. There is only one main road around the island so I’d imagine they can keep it reasonably clear. The ‘B’ roads are all gravel tracks, but much better condition that some of our A roads. If you wanna hit the F roads you have to have a 4×4.