How important is the cosmic censorship conjecture? I would like to know how important the cosmic censorship conjecture is? Should/must a quantum theory of gravity obey this? It was never rigorously proved in classical GR too. What would be the consequences if it turns out that CCC (weak) does not hold?
 A: Lehner and Pretorius have recently given some persuasive numerical evidence that there are generic violations of cosmic censorship which arise in the time evolution of black strings in 5d gravity, aka the Gregory-LaFlamme instability https://arxiv.org/abs/1006.5960. The failure of  cosmic censorship certainly does not imply the downfall of causality. Classically it means we can see a singularity without it being hidden behind a horizon. This seems to me like a very good thing. We all believe that there is some UV completion of gravity, although different camps have different views on what this is. This theory will cut off the singularity through some combination of classical modifications to Einstein gravity and quantum effects. Being able to see such a direct effect of the UV completion would be a wonderful thing, so I think we should all hope that there are also generic violations of cosmic censorship in four dimensions, although as far as I know the jury is still out on this.
A: Cosmic censorship is very important, not because it is required for classical causality (we need quantum gravity anyway), but because it is a part of black hole thermodynamics. If you believe that singularities in GR only form during gravitational collapse, the horizon cannot go through a zero temperature point, and in all the exact solutions, the extremal cases are boundaries to the naked singular cases.
The issue is confused by certain claims. First, it is important to note that censorship can only hold with matter that doesn't form domain-wall singularities in flat space. If you make dust matter with velocity directed so that all the mass ends up on a plane at some time t, you will get a jump in the metric derivative on the wall, and this is a singularity technically speaking. So you need a generic situation.
The recent claimed violation of cosmic censorship in the Gregory-Laflamme model gives you an idea of what is required. Here is a long string black hole, breaking up into spheres. The spherical black holes form, with filaments that link them, and these filaments themselves break up.
The result is that the filaments shrink to zero size, and in this limit, they vanish entirely, it is as if they were never there. This is not a violation of cosmic censorship because the spacetime in the limit is entirely smooth, each place where the line horizon once was becomes perfectly regular in the limit, except at those places (forming a dense set of points on the line)  where you have a spherical black hole.
I suppose you could consider dense set of ever smaller black holes as a singularity, but this type of thing is not in the spirit of the conjecture, since you could set this up even without the black string, just by having matter which collapses black holes at position $m/2^n$, of mass $1/8^m$ (say) where m is the power of 2 in the denominator of $m/2^n$ after you reduce the factors of 2 in m. The result is a bunch of cloaked singularities, not a singular line (at least not in the physical sense).
The issue is subtle because of these sort of scale-invariant situations, but there is no good violation of cosmic censorship today.
A: At this stage we really do not know.  There are plenty of reasons to think the CCC holds.  Classical gravity does not predict the CCC, and we must impose it.  In the thread 
Do traversable wormholes exist as solutions to string theory?
I argue this leads to some funny predictions with elementary string theory.  So it might be that string theory rules out solutions to the Einstein field equations which violate the CCC.  If this does not happen it could then be quantum gravity, in what ever form it might end up as, prevents violations of the CCC in the classical limit.  In that setting there might be quantum paths which traverse spacetime in the opposite time direction, but where these paths destructively interfere at the WKB or classical limits. Another possibility is that violations of the CCC are permitted, but where the physical solutions are simply very unstable.  So a wormhole might exist, but where the smallest perturbation send the solution into another form.  So on the solution space,or moduli, these solutions are isolated points --- give it a tweek and it moves off that solution type.
So these are not answers, but possible ways to think about the problem.  At this time we have no definative answer.
A: It's actually the case that we expect cosmic censorship to be false for a quantum theory.  
The reason why is that one consequence of classical relativity, any of the positive energy conditions, and cosmic censorship is that the area of a black hole must always increase (the so-called Area Increase Theorem).  But, if there is Hawking radiation coming from the black hole, then the area of the black hole must decrease.  If this is the case, the apparent horizon of the black hole fails to trap matter inside the black hole.  
So far, you haven't contradicted cosmic censorship--we can just interpret the Hawking radiation as a negative-energy matter distribution.  If your starting spacetime had a timelike singularity, however, as is the case for a charged or spinning black hole${}^{1}$, it will then be possible for a null ray originating from the singularity to leave the black hole interior and reach null infinity--light from the singularity will escape the hole.  This is very definitely a violation of cosmic censorship.  
${}^{1}$The picture on the right in the link is the penrose-carter diagram for the Nordstrom spacetime, which represents a black hole with a net charge.  The upward direction represents motion toward the future, and 45${}^{\circ}$ angles represent the paths of light rays.  The diamond directly below the pictured x-axis reprensents the portion of the spacetime containing a black hole (and not a white hole, which would be the diamonds above and below this one).  You can see the CC-violating path by letting the pictured horizon move upward--in this case, the 45${}^{\circ}$ rays from the bottom of the singularity will be able to leave the BH interior and reach the "ordinary space" region.
A: If the cosmic censorship conjecture does not hold, there are problems with causality in that the universe would cease to be deterministic. One could say that QM is probabilistic, but that is a misunderstanding. The Schrodinger wave equation is deterministic. Quantum mechanics is deterministic about the probabilities. General relativity is deterministic.
Situations where the CCC seem to be violated, as in the Kerr-Newman metric for spinning black holes, may not occur because of other limiting factors arising -"-Solving the above effective potential equation for the maximum under the given conditions results in a maximum potential of exactly e^2-1/2! Testing other values shows that no particle with enough angular momentum to violate the censorship conjecture would be able to enter the black hole, because they have too much angular momentum to fall in."--from Hartle's Gravity
The issue is far from settled, although, gut reaction is that it holds. (Eventhough, when asked for his gut reaction about some other thing, Carl Sagan replied "I prefer not to think with my gut." :)
