Is it possible that black holes are also neutron stars, but so dark that we cannot see them? 
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*Since the concept of the singularity in a black hole leads to infinite densities, I wonder if it is really certain that black holes exist? 

*Is there a possibility that massive objects (which are believed to be black holes) are in reality dense stars not emitting light (dark neutron stars)?

*Is there a possibility to explain the observed facts without the use of black holes?

*One more question: According to the mainstream, are neutron stars believed to
eventually collapse to a black hole, or are they stable objects?
 A: The answer is yes, but only if the mass of the unseen object is smaller than about $3M_\odot$. Generally speaking (in fact I know of no counterexample) we cannot constrain the radius of a "stellar-mass" black hole, so arguments based on density are not used.
The evidence for the existence of a black hole starts with the observation that there is a very compact, massive object that is not emitting as much light as a "normal" star of that mass.
This in itself does not rule out a neutron star, because it may well be as the OP proposes that the neutron star could be intrinsically very faint. However, "stellar-mass" black holes have been located in binary systems where observations of the orbit can give a mass estimate. If this mass estimate is larger than the maximum plausible value for a neutron star, then a black hole is usually presumed. For even the hardest feasible equations of state (the relationship between pressure and density) this is no more than 3 times the mass of the Sun. So a dark object with a mass more than 3 times the Sun is highly unlikely to be a neutron star.
A second strand of evidence comes from interactions with the "normal" star in a binary system. Neutron stars have a surface, black holes don't. Material falling onto the surface of a neutron star can cause bursts and pulsations of X-ray emission that are expected to be quite different from those produced by gas falling into a black hole.
EDIT: I add a little more detail to the second paragraph.  There is no possibility that we currently know of to support say a 5 solar mass object in any stable state, whilst at the same time accepting GR as a valid theory for strong gravity. GR itself imposes a firm upper limit, which is derived by assuming that a maximal possible equation of state, with $P= \rho c^2$ joins smoothly onto a known equation of state at lower densities. This Rhoades-Ruffini (1974) upper limit is $3.2M_{\odot}$ and can only be increased by $\sim 10$% even with maximal rotation. Thus, as these authors state in their abstract "The absolute maximum mass of a neutron star provides a decisive method of observationally distinguishing neutron stars from black holes."
It is this test (prior to the detection of gravitational waves) that makes astrophysicists confident of the detection of stellar-mass black holes.
EDIT2: There is now a further strand of available evidence in the case of merging compact objects. The gravitational wave signature of merging neutron stars and merging black holes of the same mass is, in principle different, because the neutron stars are subject to extreme tidal distortion. There may also be an electromagnetic signature, a "kilonova", from a disrupted neutron star that is not expected in the case of merging black holes. The differences occur at high gravitational wave frequencies ($>1$ kHz), to which current detects aren't very sensitive. This means that it is difficult in practice (at the moment) to use this technique.
A: 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).
A: *

*We lack direct evidence that black holes exist, but theory and indirect observations indicate that they do, e.g. there exist regions in space with gravity so strong that it curves the path of light rays significantly, the density required for compact objects that radiate high amount of energy due to inverse Compton scattering, synchrotron radiation is an indication of a black hole, density of very massive objects at the centre of galaxies etc. Infinite density is a problem of classical relativity, the quantum theory is supposed to solve this.

*The Schwarzschild radius of a neutron star is of order of 1 km, which is inside it, as neutron stars have radius of about 10 km, so if the instrument's resolution allows to discern the corresponding angles, it will be easy to distinguish between the two, if we assume that so small black holes exist. However, a neutron star cannot effect as strong gravity field as a black hole, because the latter are believed to have greater mass. Also, neutron stars do emit light, but it is faint thermal radiation.

*Regions in space with strong gravity that create gravitational lenses and the density and mass of the cores of galaxies cannot be explained in any other way, as far as I'm aware. Also, relativity has passed all the tests posed by experiments and observations so far, there doesn't seem to exist a reason to think that it's wrong in the case of black holes.

*Neutron stars are stable configurations after the collapse of a star, models indicate that black holes are more massive, therefore neutron stars cannot collapse further. Of course, this changes in the case of accretion of mass.
