I realize we're on a bit of a tangent here, but:
Re - ceremony and/or interfaith prayer with non-Natives:
Most of the Lakota I know make a pretty firm distinction between prayer and ceremony. Most folks I know consider prayer something anyone can do, especially if you do it in the language of your own ancestors. Whereas ceremony usually refers to the Seven Sacred Rites. However, non-Natives usually have no idea there is any kind of distinction there, and are unclear on the terminology. So, if a non-Native describes something as ceremony, I'd ask the Natives that were there what really happened before calling it ceremony.
That said, I have participated in interfaith prayer events that were set up by Lakotas in a Lakota framework, and songs in Lakota were sung, along with making space for people of other cultures to also offer prayers and songs. What I saw was that, if culturally rootless non-Natives were allowed to witness how the space was set up, and the building of the sacred fire, and allowed to listen to the traditional songs, and see how offerings were made, they left that "prayer only" event to mimic these ways in their own lives, and even to sell them to other non-Natives as sacred Lakota ceremony.
Everyone offering prayer in their own language, and building the fire in an interfaith manner is one thing, in my non-Native opinion. But as soon as any degree of traditional ceremony is included, even if it's "just" the placing of flags, prayer ties, smudging, drumming and using sacred songs, the appropriators are there to steal it.
It's one thing when interfaith events are just prayer, or when the only non-Natives who are invited are trusted relatives, but any time these things are advertised and open to the public, they are now overrun with appropriators, there to data mine. My area is now overrun with pretendians, some of whom got their start by watching and imitating NDNs at these events.