The Copenhagen interpretation by Niels Bohr insists that quantum systems do not exist independently of the measuring apparatus but only comes into being by the process of measurement itself. It is only through the apparatus that anything can be said about the system. By necessity, the apparatus has to be outside the system. An open quantum system. Can quantum mechanics be applied to closed systems where the measuring apparatus is itself part of the system? Can a measuring apparatus measure itself and bring itself into existence?
''that quantum systems do not exist independently of the measuring apparatus but only comes into being by the process of measurement itself'' is a gross distortion of the Copenhagen interpretation. The latter only asserts that the particular value of measuring quantum variables of a system that exists objectively (otherwise how could it be measured) is predictable only within its intrinsic uncertainty.
The solar system is a quantum system whose state we know reasonably well in a coarse approximation appropriate to such big systems, as we know its thermal properties and quantum gravity effects play no role. All our experiments so far have been performed within this quantum system, and all our measuring instruments are part of it.
Every individual measurement we do is in fact a measurement of the state of a tiny subsystem, sometimes (spin or polarization measurement) of only a single quantum degree of freedom, and thus reveals a tiny little bit more about the state of the solar system, namely about the substate obtained by tracing out all other degrees of freedom. This tracing out is the source of decoherence, which is frequently well approximated by the Copenhagen collapse postulate.
Thus there is not the slightest trace of the mystery the OP seems to suggest.
Q: Can quantum mechanics be applied to closed systems where the measuring apparatus is itself part of the system?
A: Definitely, yes. Sometimes it is even necessary, such as in the case when you place an atom ("system") in between two mirrors ("apparatus"). The resulting quantum mechanical model is that of cavity quantum electrodynamics. Now if you want to know what's going on there, you have to bring in a second measuring apparatus (a photon detector, say).
Q: Can a measuring apparatus measure itself and bring itself into existence?
By definition, the "apparatus" is the thing doing the measuring. The "system" is the phenomena under investigation. If you choose make the object of interest system+apparatus then you've just redefined what the "system" is and you are going to need a new apparatus to do the "measuring". So, if you believe the sentence "It is only through the apparatus that anything can be said about the system", then the answer is no.
This is the basis for the Many-Worlds interpretation, and many-minds/decoherence/consistent-histories variations. The basic point is that you can consider quantum mechanics as a complete description of nature, but only at the cost of a nontrivial indentifications of states of memory of an observer with the system. This is discussed well in many places, originally, "The Many-Worlds Interpretation" from 1972(?), edited by DeWitt, reprints Everett's original thesis, which has many interesting results.
I have heard it said that the Copenhagen interpretation has been by and large abandoned by 'real' physicists...maybe it was Weinberg who said that in print, and maybe it was in his generally excellent book, Towards a Final Theory. Hannabuss told me, fifteen years ago, that the time is long past for axiomatic gassing and philosophising about measurement, it is now time to analyse it as a concrete physical process with a Hamiltonian and everything (of course, using whatever approximations are needed to get some sort of answer, he himself has an analysis of the famous Dirac polariser argument but with a semi-classical approximation for part of it, so this is all still work in progress). And many other excellent physicists have done this, or at least started it, as well. See the references to Collet, Millburn, Walls, and also Gardiner and Zoller, and Allahverdyan, Balian (long-time head of one of the theory divisions at Saclay and called by Streater one of the most interesting theoreticians alive, see
pointing the way to the future of Physics, which BTW is Statistical Mechanics) and others in my own Thermodynamic Limits, Non-Commutative Probability, and Quantum Entanglement
and Hilbert's Sixth Problem, the Axiomatization of Physics
That is to say, such physicists are indeed analysing the combination of microscopic system being measured with the macroscopic measurement apparatus as a closed system obeying a joint Hamiltonian made up out of the individual Hamiltonians and an interaction term and obeying the laws of the unitary evolution of linear Quantum Mechanics. Perhaps not many would agree that they have 'solved' the Quantum Measurement Problem, but some of them think so, and it has to be admitted that there are sufficiently many degrees of freedom inside the measurement apparatus that one could imagine decoherence going on in this closed system, so it is consistent with the rather different decoherence crowd, but much more physically based.
I do think they are on the right track, I hope you will try to look at their stuff, but some of it predates the Los Alamos free archive, still the Balian stuff is very recent Armen E. Allahverdyan, Roger Balian and Theo M. Nieuwenhuizen at arXiv:1003.0453
I myself feel that although their physics is more or less correct, they are logically circular and axiomatically sloppy. Also, their model although more realistic is not different in principle from the much quoted early work of H.S. Green.
No, it cannot. The observer cannot determine his own quantum state and as such he does not obey the usual laws of quantum mechanics.
In this sense quantum mechanics is not an universally valid theory.
See this paper for a proof