In QED, each vertex has 3 lines attached to it: 2 electron/positron lines and one photon line.

But in other quantum field theories, there are vertices with more edges. In QCD, for example, there are vertices where 4 gluons meet. In the Standard Model, there are vertices where 4 Higgs particles meet, and vertices where 4 electroweak bosons meet, and so on.

The numbers 3 & 4 aren't really special. What's special is the numbers we assign to the edges. In 4 spacetime dimensions, any gauge boson (like the photon or gluon) gets the number 1. So does any scalar field like the Higgs. Fermions like electrons and quarks get the number 3/2.

These numbers are important because you should never meet a Feynman diagram in a particle physics computation where the sum of these numbers at any vertex is larger than the spacetime dimension 4. This is a fairly deep fact about physics. Diagrams that violate these conditions are strongly suppressed at "low" energies (such as those accessible at the LHC). They contribute almost nothing to the Feynman diagram sum.

Edit: This is Wilson's explanation of renormalizability, adapted to Feynman sums. It's the explanation for why physics is possible: The details of short distance physics (think Planck scale) can be very complicated, maybe involving lots of complicated Feynman diagrams. But at low energies, almost all of that complication averages out, and we can adequately approximate the laws of physics using simple ingredients.

Those labels seem related to the spin, is that right?

Indirectly right. Those numbers are the scaling dimensions of the field operators. (Rescale your ruler by a factor f, and the fields will get multiplied by f to the scaling dimension.) These numbers are fixed by requiring that the kinetic energy density transform correctly under scaling. The form of the kinetic energy, in turn, depends on the spin of the field quanta (i.e., the particles).

Feynman diagrams are a series expansion, not unlike Taylor series. We do it for things which cannot be computed exactly. (I don't want to go if it is more "Nature", "our description of Nature" or "a box of numerical tricks we have mastered"... and frankly, I don't believe there is a strong distinction between these three.)

3 lines stand for the lowest non-trivial order (like 'x' in a Taylor series; '1' would be a line) - more complicated things can be made out of these.

I recommend the first two chapters of Griffiths's Introduction to Elementary Particles which gives a very readable approach of showing which Feynman diagrams occur and what they mean. After that, the book gets more technical, but the first two chpaters give historical background and make it feel like playing with tinker-toys where you can just snap together various basic diagrams.

I heard a good rule of thumb to follow when drawing Feynman Diagrams is that the fewer lines there are, the more probable the occurrence of the event represented by the diagram. So at the simplest case, a single line representing an unchanging particle, is far far more likely than any two particle interactions (3 lines) and so on.

This is true if the system you're trying to describe is weakly coupled.

If the theory is strongly coupled, then in fact more and more complicated diagrams count more and more. In this case the method of Feynman diagrams because basically useless---in the case you describe you know you can get most of the answer from a simple calculation (of the simplest diagrams). But if more complicated diagrams count more (as in strong coupling) then you don't have a place to start, because whatever diagram you pick to start at I can make more complicated and be confident that my new diagram counts more than the pieces you've computed.

In that case we need a different approach, the most generic of which is lattice field theory

https://en.wikipedia.org/wiki/Coupling_constant#Weak_and_str...

https://en.wikipedia.org/wiki/Lattice_field_theory

Roughly, though it is actually the number of vertices and depends on the particles involved (obviously a diagram representing something very unlikely is less likely than some normal process)

Feynman's Diagram are just abstraction that manages to abstract away some elements of nature into a simplified version.

Fetishizing them is like fetishizing the way Ancient Egyptians wrote their fractional numbers.

That doesn't say much though.

Something that is "just abstraction" can still be world-changing...

In fact those are the most world-changing things (abstractions), the concrete stuff mostly derives from there...

If you focus too much much on current abstraction, you lose sight of better ones - that was the point of my previous post.

I also don't agree concrete stuff derives from abstractions. You can make concrete stuff, using certain models/abstractions, but their behavior will always differ from the model. Because model aren't reality.

What exactly is it that it says about nature?

QED!

QED as in Quantum Electro Dynamics? :)