I don't think there is a particular schedule. They are not announced so it is hard to tell. It seems that if there is a new, themed event or a new series of Divines that there are most likely to be new Rays to match it. There is probably someone who keeps better records than I do.

For the item itself: It's not very often, think last time it was in an Event if I remember correctly.

The designs: Pretty much every event that has a new background, can be found in the Ow's H Ray.