Their angular momentum stays nearly constant. You might be thinking of their angular velocity.
There is a lot of simulation-based work out there on the inspiral of two compact supernova remnants (NSs or BHs), done partly to determine what the gravitational wave signals would look like for the LIGO experiment. Two NSs would merge to form a BH, because their combined mass would be above the threshold for the fermi pressure of the NS to resist gravitational collapse. Part of the NS material would briefly orbit the newly formed BH as an accretion disk, and some of the energy would be beamed out along the rotational axis, visible to us (if we happen to be in the line-of-sight of the axis) as a gamma ray burst.
A popular assumption about black holes is that their gravity grows beyond any limit so it beats all repulsive forces and the matter collapses into a singularity. [...] Is there any evidence for this assumption?
It's not an assumption, it's a calculation plus a theorem, the Penrose singularity theorem.
The calculation is the Tolman-Oppenheimer-Volkoff limit on the mass of a neutron star, which is about 1.5 to 3 solar masses. There is quite a big range of uncertainty because of uncertainties about the nuclear physics involved under these extreme conditions, but it's not really in doubt that there is such a limit and that it's in this neighborhood. It's conceivable that there are stable objects that are more compact than a neutron star but are not black holes. There are various speculative ideas -- black stars,
gravastars, quark stars, boson stars, Q-balls, and electroweak stars. However, all of these forms of matter would also have some limiting mass before they would collapse as well, and observational evidence is that stars with masses of about 3-20 solar masses really do collapse to the point where they can't be any stable form of matter.
The Penrose singularity theorem says that once an object collapses past a certain point, a singularity has to form. Technically, it says that if you have something called a trapped lightlike surface, there has to be a singularity somewhere in the spacetime. This theorem is important because mass limits like the Tolman-Oppenheimer-Volkoff limit assume static equilibrium. In a dynamical system like a globular cluster, the generic situation in Newtonian gravity is that things don't collapse in the center. They tend to swing past, the same way a comet swings past the sun, and in fact there is an angular momentum barrier that makes collapse to a point impossible. The Penrose singularity theorem tells us that general relativity behaves qualitatively differently from Newtonian gravity for strong gravitational fields, and collapse to a singularity is in some sense a generic outcome. The singularity theorem also tells us that we can't just keep on discovering more and more dense forms of stable matter; beyond a certain density, a trapped lightlike surface forms, and then it's guaranteed to form a singularity.
Why can't some black holes be just bigger neutron stars with bigger gravity with no substantial difference except for preventing light to escape?
This question amounts to asking why we can't have a black-hole event horizon without a singularity. This is ruled out by the black hole no-hair theorems, assuming that the resulting system settles down at some point (technically the assumption is that the spacetime is stationary). Basically, the no-hair theorems say that if an object has a certain type of event horizon, and if it's settled down, it has to be a black hole, and can differ from other black holes in only three ways: its mass, angular momentum, and electric charge. These well-classified types all have singularities.
Of course these theorems are proved within general relativity. In a theory of quantum gravity, probably something else happens when the collapse reaches the Planck scale.
Observationally, we see objects such as Sagittarius A* that don't emit their own light, have big masses, and are far too compact to be any stable form of matter with that mass. This strongly supports the validity of the above calculations and theorems. Even stronger support will come if we can directly image Sagittarius A* with enough magnification to resolve its event horizon. This may happen within 10 years or so.
Best Answer
Let me start with your question about stability: Any astrophysical object is subject to a battle between two forces: gravity (which will try to collapse the object) and whatever force prevents that collapse. A regular star uses heat (generated by thermonuclear fusion) to counteract gravity. When it runs out of fuel, gravity begins to compress the star further. Here are three different possible end states: a white dwarf, where the degeneracy pressure between electrons (that is, the Pauli exclusion principle as applied to electrons) is sufficient to balance gravity; a neutron star, where the degeneracy pressure between neutrons (that is, the Pauli exclusion principle as applied to neutrons) is sufficient to balance gravity; or a black hole, where there is no force/pressure that is strong enough to counteract gravity, and all the matter collapses (in classical GR, to a point/singularity) under gravity. The white dwarf and neutron star are stable unless they grab too much mass from somewhere else.
As for the rest of your question, it depends on what you mean by a black hole. Are there regions of space from which no light can escape (trapping horizons)?: almost certainly. Supermassive black holes can have large horizons, with surprisingly small space-time curvature, and we understand gravity and GR well enough that we can be reasonably sure that such horizons exist. Are there singularities inside these horizons? - almost certainly not. Physicists dislike singularities, which is one reason why they search for a quantum theory of gravity. So the question of what lies inside a black hole can only be answered when someone comes up with a consistent quantum theory of gravity.
We know enough about electron physics to suggest that there is a limit (the Chandrasekhar limit) to how massive a white dwarf can get, and similarly we know enough about neutron physics to suggest that there is a limit (the Tolman–Oppenheimer–Volkoff limit) to how massive a neutron star can get. Beyond this our knowledge of states of matter is shaky, so yes, there could be a quark star or some other exotic state of matter whose degeneracy pressure can counteract gravity. But the general trend is that there is a limit to such forces, and that for a sufficiently massive object there is no way to stop complete gravitational collapse.
Observational evidence for black holes typically comes down to: we know that there is a massive object in this region of space (by looking at the objects that orbit around it), and we know that it packed into a volume of space that is at least this small (by looking at accretion disk data, for example). The density we compute from that mass and volume is too high for a neutron star, so in the absence of evidence for various exotic stars/states of matter, we shall assume it is a black hole.
EDIT: As noted by the commenters below, the density (Mass over Volume) for black holes can be quite low; it is more accurate to say that the Mass to Radius ratio becomes too high for it to be anything but a black hole (i.e. all the mass is contained within the Schwarzschild radius, and so it undergoes gravitational collapse).