This is a member of the Nidulariaceae / birds nest fungi, the exact variety would need better photos and possibly a microscope, but I can still answer your other questions:
No, they are not dangerous. Inedible, but they don't damage your garden. They do what they do best, which is eating the decomposing wood from your mulch.
The white "felted" disc underneath is the actual fungus, the mycelium, the cups with the lentil-shaped peridioles are only the fruiting bodies. Think of the mycelium as the apple tree and the fruiting bodies (which we often call "mushroom") as the apples which contain the seeds (spores).
For removal you can simply lift the mycelium, preferably before the cups open. Another option would be to keep your mulch very dry (probably not feasible). But I wonder why - if you spread mulch it's normal for it to break down and fungi are a natural part of this process. If they appear, your mulch has already started to decompose. In this case you would have to regularly (e.g. annually) excange the "old" mulch for fresh mulch, but apart from esthetical reasons (and beauty is in the eye of the beholder, I personally love to see nature do it's thing where possible) I see no necessity to do so, especially if your neighbour has "a larger patch", as you wrote, meaning you will constantly get new spores into your mulch.