A shelter doesn't have to be very big, 10x10 would be great. Just an overhang off an existing building would work. I have used straw as bedding when my girls foaled, they mostly sleep out in the open so I use nothing now. The problem with straw is it is more difficult to keep clean and your going to use more. If your grandpa has lots of it, go ahead and use it. One good thing about straw is its breaks down in a manure compost pile better than wood shavings.
As far as the wire, can you just remove it?
Sounds like you have a good deal going if you can keep your horse across the street. Plus I'm sure your grandparents will be thrilled to see more of you too.
"Until one has loved an animal, part of one's soul remains unawakened..."
- Anatole France