My understanding is that QM tunneling tells us how likely it is that a photon, electron, etc. will make it through a barrier. In layman's terms, is this true/correct?
I can offer a simple example, which can hopefully give you some idea. I will try to be more qualitative to capture the key idea. Before that, note that think of a 'barrier' as a finite energy barrier, not as a physical 'wall'. If we do have an idealised 'wall', which we may model as an infinite potential, then even quantum tunnelling won't get us through.
The example that I will give is a classic one - nuclear fusion of hydrogen into helium in the core of the Sun:
Since the nucleus of a hydrogen atom is positively charged, to fuse two hydrogen nuclei into a helium nucleus requires a lot of energy (achieved by high temperatures) to overcome the electrostatic repulsion. This requires a temperature that is higher than that in the core of the Sun. This means that without quantum tunnelling, nuclear fusion simply won't happen at a rate rapid enough to support the Sun (More precisely, fusion will still happen for the hydrogen atoms that are moving faster than most others, i.e. those in the "Maxwell-Boltzmann tail"). For an average hydrogen pair, we can think of this electrostatic repulsion as an energy barrier preventing fusion to happen.
Now comes quantum tunnelling: note the 'barrier' of this interaction is finite, i.e. the energy required for fusion is not infinite. What this means is that it is possible that some hydrogen nuclei, which do not possess sufficient energy classically, can have enough energy to fuse (c.f. Heisenberg Uncertainty Principle). Since there are a lot of hydrogen atoms in the core of the Sun, despite the low probability of this process, a considerable number of hydrogen nuclei are still able to quantum-tunnel and fuse (per second). The end result is that the Sun can support itself and I can answer your question here.
I hope I didn't go too off-topic. I think this is quite a nice application of quantum tunnelling and illustrates some of the key points.
I'll try to answer to all three points, if I have correctly understand you questions.
QM is about wavefunctions. One fundamental propriety of a wavefunction is that it is normalized: it means that its integral all over the space must be equal to one. This leads to the fact that the squared module of the wavefunction is the probability distribution for the position of that particle. I think that a possible, intuitive solution for this question comes from the problem of a particle in a box. For a 1D cases, you can draw the probability function for position for increasing energy (as it is done here in Trimok's answer) and see that it has nodes (i.e., points where it is null) every time nearer. This oscillations are fundamentally connected to the quantum behavior of that particle: you can only observe that their mean value reduces to the classically expected one (as done in the particle in a box example, where the average probability density function is greater near to the walls).
Quantum Mechanics, through some approximations, can be used to calculate ab initio the structure of molecules and their macroscopic proprieties (see my answer here for a brief description of how to describe bonds in a molecule). For instance, you can calculate the spacing of the atoms in a certain molecule or if it is paramagnetic.
Yes, I think this is a correct statement. You can actually calculate the probability of that tunneling event, as done here for some barriers.
The original answer was for a different question that had three different points; the answer to the actual question is what I called point 3.