Yes the clay would be covered with lime, at least a finish coat to seal it all off.

I wonder the same about the foam. I don't think it should be a major problem though as it does not seem like water would make it that far.
I would be worried though about the foam holding water there in the case that moisture does penetrate that far.
If it is a major concern a heavy clay could be plastered on the inside in stead, but wouldn't be quite as air tight. Remember that the whole reason for the spray foam is to stop drafts. As such it COULD be considered an extra feature and is not absolutely necessary.

Right now I am trying to work out in my mind how the system would be executed around braces. One drawback is that this all has to be put in place as the frame goes up, before horizontal timbers such as plates top off the cavities. order of assembly suddenly becomes very important!