Just from reading about how to keep them (freshwater jellyfish)intentionally it seems they are somewhere between hard and impossible. The linked article from a post above says none of the public aquariums that have tried have been successful and that Wheaton college maybe has figured out how to do it. I would think if you crank up your filter, they will find their way in and die as soon as they are capable of free swimming. This would explain why the couple posts where someone else thinks they have had them in the past, they simply disappeared.
I can't confirm if you have jellyfish from the pictures as I'm not experienced enough to rule out the multitude of possible fungus that exist but if it were the jelly fish, maybe it's not as problematic as it seems.
Upon reading this thread tonight, I learned about something I was totally unaware of, and I thank all you guys for sharing, and I thank OP for posting the pics.
Anyone have a spare round tank with a custom filtration system and tons of live microfood? I would love to see a fellow hobbyist grow one of these out to full adult just to say they could do it. I would try but I don't think I have the skill or space.