Carbon structures, fiberglass structures, and other composite matrix structures aren't the only materials that can look fine on the surface yet have trouble out of sight. The most basic building material, the one we've been using in airplanes long before metal and glass, is wood...and that can appear perfecton the surface but be completely rotted beneath. Further, it doesn't lend to ultrasonic detection or other forms of nondestructive testing that we take for granted on most structures today.
The traditional method for testing glass structures, believe it or not, is the coin-tap test. Tapping with the edge of a coin, one listens for a change in the sound. It doesn't sound very high tech, but it's also very effective.
Glass structures don't lend well to many types of NDT testing, either. However, this doesn't mean that they can't be tested, or maintained.
Heat and vacum bagging and pressure moulding are various methods of establishing a cured part in a particular form or shape, but in the end the result is the same; a compromise between the advantages and weakneses of the matrix and the cloth or thread fiber balanced agaisnt stiffness, flexibility, fatigue resistance, weight, and overalll strength when considering the complexity of the part.
One of my flight jobs involves wearing a helmet. I had mine custom constructed by a well known helmet manufacturer using kevlar, instead of their usual glass layup. I elected to go with the added expense for increased strength and puncture protection...and a couple of years ago I got to put it to the test. All I'll say about that was that it was worth the investment.
Where a glass mat surfboard might blow apart under enough stress, the same isn't true of many fiberglass structures or carbon fiber (kevlar, spectra, etc) structures. In fact, often they remain surprisingly intact. You might be surprised at the resistance to failure of a composite part vs. a typical metal semi-monocoque structure that we see in an aluminum airplane.
You may be surprised too, at the amount of glass and composite structures flying around today...which do very well indeed.
So far as what happens in a crash, well...far too many variables are involved. We can design a structure to withstand known or expected inflight loads, based on load paths and values. We can't begin to predict what may happen to a structure during a crash because we have absolutely no idea what may be done to the structure during the crash, or where the loads may be imposed. A structure may be extremely strong in tension, but not compression...and a compression load, or bending load, or shear load, may cause it to fail very quickly.
We recently saw the destruction of a B747 in a crash; the flight deck remained intact where the remainder of the structure was predictably destroyed. I'd have bet such a thing would never have happened; I was as surprised as anyone at the fact that not only did the flight deck remain intact, but the eight souls on board survived. That was a metal airplane, of course. What would have happened had it been carbon fiber? We can't possibly say, but I would suspect that the integrity of the structure would certainly fare as well, if not better.
There are so many directions the discussion could be taken ranging from flame propogation to the ability or inability of rescuers to cut through the structure during an extrication. All valid issues, but the question you posed was one of airworthiness. Certainly if the structure is designed properly and aerodynamically, composite structures have been key parts of airplanes for many decades now, and will continue far into the future.
Carbon fiber is just one more construction material; something to be learned, but not feared. It's used because it's advantages outeigh it's disadvantages. Metal structures fatigue, and have their own disadvantages, but also have advantages. Same for glass and carbon. Just another material, with some fantastic possibilities.