I think you got it exactly backwards. There are theoretical demonstrations of the Einstein-Rosen bridge, i.e. you can write down the Schwarzschild solution the Einstein field equations, do a couple changes of coordinates, and demonstrate the system contains a wormhole; see any general relativity textbook for this. There are of course no experimental demonstrations -- that would be sensational.
The other objection you have is that singularities should be impossible because the universe is expanding. This doesn't make sense on the scale of a black hole because the expansion of the universe is very weak; it's totally undetectable for anything short of cosmological scales.
On the other hand, it's a fair question why the universe has concentrated lumps of matter in it, such as galaxies and stars, when you would expect a uniform distribution. This is due to gravitational instability: matter attracts matter in a runaway process, so any inhomogeneities get amplified. Remarkably, this doesn't violate the second law of thermodynamics because lots of entropy is produced by gravitational potential energy going into kinetic energy during collapse. (I've heard some people go so far to say that's the fundamental reason life can exist at all, though it really depends on what you mean by 'fundamental'.)
Mathematically, once you have these concentrated lumps of matter you can prove sufficiently dense matter must form black holes, e.g. the TOV limit for neutron stars. Then the Penrose-Hawking singularity theorems essentially say that if gravity in a region is strong enough, such as in a black hole, there must be a singularity.