Do these use up a breeding point?  I never paid attention and was curious.  I checked the Desert Biome guide and only see it mentions the vermillion reward, but not if a breeding point is used.

It doesn't use up any points :). You can check this pet page: http://beastkeeper.com/pet/484201 - she recently had one child (from the shrine) and still has all breedings unused (0/23).

Thanks adgin.  I know I had used them before but honestly never thought to look at the breedings before/after.