Hi Susie, as soon as I can get to town I was going to see if there is some type of glue that would stick on Styrofoam, but I was thinking more of stacking layers of it to make it stronger. How about if you cut off strips from the edges of the board, and randomly glue them on top of each other, then when you use the wire brush it would blend the edges together, I would only go maybe two layers thick for the sides. Does this make sense?
For your box, if your babies have come from somewhere cold, they will be fine outside, in fact that would be better. How about if you put your box outside, plant it there. You can cut the sides down to make a shallow tray too, so it won't be heavy.
