"Does the type of backing matter as to when it is applied?"
Yes, it does.
Most of the time, somewhere early in the tillering process is a good time. A wooden backing often needs to be applied earlier, as the backing will seriously effect tiller and increase drawweight. A rawhide backing is often applied at or right after floor tillering.
A light backing that is mainly for looks (snakeskin, silk, feather, bark...) is usually applied at the very end, when tillering is finished.