Is ch. 2, sect. 4 of The Feynman Lectures on Physics, Vol. 1 still accurate? The chapter 2 section 4 of volume 1 is on nuclei and particles.
Here are a few things that trouble me. Dr. Feynman says that

Another most interesting change in the ideas and philosophy of science
  brought about by quantum mechanics is this: it is not possible to predict exactly
  what will happen in any circumstance. For example, it is possible to arrange an
  atom which is ready to emit light, and we can measure when it has emitted light
  by picking up a photon particle, which we shall describe shortly. We cannot,
  however, predict when it is going to emit the light or, with several atoms, which
  one is going to. You may say that this is because there are some internal "wheels"
  which we have not looked at closely enough. No, there are no internal wheels;
  nature, as we understand it today, behaves in such a way that it is fundamentally
  impossible to make a precise prediction of exactly what will happen in a given
  experiment.

Has this statement changed at all? He says that there are no internal wheels. At one point I also remember him mentioning that we do not know what is going on inside a nucleus. Is this still true?
One more extract is this:

just like with a field of interaction between charges and photons, we made quantumelectrodynamics, with a field of interaction between neutrons and photons, we can make quantum'NUCLEO'dynamics 

He later mentioned that the calculations are going on for 20 years and are too complicated to solve.
What about this theory? Has it been proven or disproven? Or are we still not capable of calculating it?
The last thing that bothers me with this chapter is this:

So we are stuck with a theory [he is talking about the pion and mesons etc.], and we do not know whether it is right or wrong,
  but we do know that it is a little wrong, or at least incomplete. While we have
  been dawdling around theoretically, trying to calculate the consequences of this
  theory, the experimentalists have been discovering some things. For example,
  they had already discovered this m-meson or muon, and we do not yet know where
  it fits. Also, in cosmic rays, a large number of other "extra" particles were found.
  It turns out that today we have approximately thirty particles, and it is very
  difficult to understand the relationships of all these particles, and what nature, wants them for, or what the connections are from one to another.

Is this condition still the same? Have we tied the loose ends together yet? Or do we still not know what these particles do?
What is the prevailing theory currently about these particles?
 A: The first quote that Feynman wrote in the 1960s remained as accurate and crisp as it was 50 years ago. The quantum randomness can't ever be predicted, even in principle, so there are no hidden wheels here (which we would call "hidden variables"). The proof that hidden theories couldn't be right were more or less available during the publication of Feynman's textbook but they became much more solid in the following decades. At any rate, top physicists knew the right answer – the quantum randomness is here to stay and can't be "reduced" to anything non-random – from the beginning i.e. from the mid 1920s. Nothing will ever change about the impossibility of "hidden wheels" in physics.
On the other hand, the second quote about the nuclei is obsolete because we indeed have a more refined theory of the interior of protons and neutrons, the so-called Quantum Chromodynamics (QCD), that was found in the early 1970s, about 10 years after the publication of Feynman lectures. This new theory builds on the so-called quarks that are "colorful" (a type of charge) and therefore interact with a new force (strong force) that is mediated by the "gluons". At short distances, the strong force is weak (asymptotic freedom) but it becomes very strong and "confining" (making it impossible to isolate colored objects altogether) at long distances.
This QCD, together with much of the rest of the so-called Standard Model (the electroweak theory with the Higgs mechanism), was added to physics 5-10 years after the Feynman lectures, so Feynman's description about the incompleteness of the "current theory" at that time was very reasonable, too. In that time, Feynman would talk about the "state-of-the-art theory" by which he meant QED (Quantum Electrodynamics) with protons and neutrons added as elementary charged particles and with the non-renormalizable four-fermion Feynman-Gell-Mann interaction added to account for the weak nuclear force. Of course that we know that such a mixed theory was inconsistent at high enough energies and failed to reproduce many properties of the strongly interacting particles (hadrons).
A: A brief complement to Lubos' very clear answer:
Not only does the fundamental randomness associated with quantum projective measurements still hold theoretically, but state-of-the-art experiments have now managed to demonstrate it, directly, on quantum systems like


*

*single atomic ions in traps,

*single photons in cavities,

*collective mechanical oscillation modes of small macroscopic oscillators,


and a large list of others. It is possible to make a quantum 'cat' that can be 'dead' and/or 'alive', prepare it in a superposition of 'dead' and 'alive', and experimentally observe it to be 'dead' or 'alive' with equal likelihood. Quantum jumps, for example, can easily be observed now: an ion will either fluoresce or not, and discretely jump between the two states, but cannot be in an in-between state.
