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Sara, I use the gray grit, it has pieces of charcoal and oyster shell already in it. Sometimes I will use the red grit if my store is out of the the gray grit and I am totally out.
You could move them now, before the chick is hatched, they may or may not abandon the egg, but I rather doubt it. If you move them now, move the male first so he can inspect the nest site. As to moving them once the chick is hatched, I would wait a few days afterwards. As these are the only two that you have then it probably would be safe to move them to the new location, just make sure you move the hen and chick together and place them in the nest first and then bring the male over. Move the material in the current nest with them so that the new nest is familiar to them. Pigeons like to pick the highest nest area they can, and they are not the greatest nest builders either.
As to the male doing what he is doing when you coo to him, he is either looking at you as a potential mate, even though he already has one at "home" with a baby, or he is trying to drive you away from the nesting area. I tend to lean towards the first explanation more than the second. While they tend to be somewhat monogamous, a male is not above chasing after another female that is apparently available.