Is there, within halacha, any prescribed period of time that MUST pass before a headstone can be placed / unveiled? If not, can it be be in place on the day of the funeral ?

According to Kabbalah the headstone shouldn't be put during the seven days of mourning, but it is good to attend to it right after that so that it will be up as quick as possible. It is custom to carve the name close to the Shloshim (30th day), and to those who have the custom of visiting the grave on the Shloshim, have the unveiling then. Rabbi Ro'i Margalit