In a couple of years I hope to cross Iceland east to west through the centre. I understand the the roads in central Iceland are only open in the summer. Is that for vehicles only or for walking as well? I looked around but most info is about (4x4) cars and other vehicles.
Since Iceland has the "right to roam" similar to other Nordic countries, you're virtually allowed to go wherever you want. You probably won't even be able to see the road due to the snowcover, so it would be weird if you could go everywhere except there :) so I wouldn't worry.
As for rescuing, that's technically a different question. That should just be a matter of insurance.
Hiking in the Iceland highlands/roads when there's snow on the ground is certainly allowed, although you'd find travel much easier on skis. Be aware in the winter/spring the roads and marker / sign posts are totally buried in snow.
The mountain rescue service in Iceland will be happy to rescue you should you get into trouble - register with ICESAR in Reykjavik before you head out (They will also rent you a PLB if you want
There's no need for rescue insurance in Iceland, they don't charge for rescue.