so, I recently bought a Lunette cup (size 2). Had some insertion issues, but got over them with a different folding technique (they list them on their website and even have a menstrual cup origami video link). The placement was nice, no pain, no leaks, just over the pubic bone. Then, after a workday, I tried to remove it. I spent 2h in the bathroom trying different methods, but finally gave up and went to sleep.
Now, I'm writing this, because during my struggles I read about similar issues on the internet, which helped me not lose hope. And there's a happy ending. This morning I took a muscle relaxant (don't know if it helped any), and pushed my index and middle finger alongside the cup. What seemed to work best was starting while lying down during insertion and then coming up to a sitting position to make myself shorter (this all took place in the bathtub) and just having the fingers in there as far as I could reach and pressing on the cup until I could break the seal to end all seals. After that I could slide the blasted thing out.
Stuff that didn't work, and considering the force it took to break the seal, couldn't work:
Pulling on the stem and trying to move the cup back and forth;
Grabbing the bottom of the cup and pinching and pulling (first of all, I couldn't get a hold of the cup bottom, it's tight in there, okay; secondly, just trying to do it with this cup outside of the body didn't change the shape of the upper part enough to have any effect on the seal; thirdly, without breaking the seal it would be impossible to generate enough force in that awkward hand position to move the cup);
Bearing down (was completely useless, the only useful thing to do is try to find a position where you're shorter, if you're having as much problems as I was that thing is stuck to your vaginal wall like a baby octopus and it's not coming lower unless the actual wall is coming lower).
All in all, not feeling terribly empowered or enthusiastic about menstrual cups, but I hope this helps someone.