I don't think there are actual rules about veils -- it seems to be personal preference, tradition and culture. For instance, the on/off face issue. I was looking at videos of the weddings of the Infantas of Spain. Elena wore her veil over her face, and Christina wore it pushed back. In the UK the royal brides tend to wear the veil off the face so that the bride can be seen, but in Diana's case (and Sarah's) she wore it over the face until they went to sign the register and she came back with it re-arranged.
The origin of the material of the lace. They seem to be blessed with antique lace belonging to grannies and great-grannies, so it would be a nod to tradition and a remembrance of a grandmother who's probably gone now.
Length? Probably dependent on the lace available to the brides and their dress designers.
Tiaras probably not required -- but if you have one or the groom's family has one, you probably won't get many opportunities aside from the wedding to wear one, so why not? Princess Caroline of Monaco didn't wear one at her first wedding (nor at the others, for that matter). Sarah Ferguson started out with a garland of flowers and returned with the tiara after her veil had been moved off the face.