I believe the article’s point was that there is no “they do”, “they believe”, or “they” anything else as there is no one single group involved. There are a number of groups that go by that name, some related and some not. I imagine it would not be possible to summarise all the beliefs and positions in the space allowed in the article.
Bigger Secrets by William Poundstone has a fairly in-depth essay on the current (more or less) goings-on in the Rosicrucian order regarding rituals, initiation, etc. They seem relatively innocuous as far as “secret” societies go.
I’ve been to the Rosicrucian place in San Jose; it’s worth a visit if you live near there. It’s…interesting, if wacky. There’s a museum with replicas of ancient Egyptian stuff, ‘the tower of Babel,’ and so on. I seem to recall that there’s a planetarium too, but I may be wrong about that. There are statues and gardens of papyrus and mystic gates. (My dad amused himself by taking pictures of a hugely pregnant me standing next to Toueris.)
Their beliefs are fuzzy, but seem to be about ancient Egyptian wisdom coming down through the centuries, I think by way of Templars and secret societies and so on.
The only reason I didn’t go too deeply into Rosicrucian “beliefs” is that because the original group (if there ever really was an original group) didn’t have “beliefs” as much as they had “ideals” which were pretty much in tune with the Enlightenment and what they called the “general reformation of the age.” It really wasn’t too different from what Francis Bacon was proposing except it had the addition of a strong alchemical element.
As far as what was happening at Notre Dame, I can’t say. Since Rosicrucianism was not a religion, they did not have priests so the group meeting there was most likely Rosicrcrucian in name only and had not connection to the original group; a status that I believe applies to any group that calls itself Rosicrucian nowadays.