For instance, how did he come up with interpreting the propagator as the propagation of particles?
The path integral is usually introduced as a matrix element of the time evolution operator
$$
\langle x_f\lvert\mathrm e^{-\frac{\mathrm i}{\hbar}\hat{H}(t_f-t_i)}\lvert x_i\rangle,
$$
which is a measure of the probability of finding a system in final state and time $x_f,t_f$ when it had been in state $x_i$ at time $t_i$ initially. It is quite plausible to name it propagator as it gives immediate access to the probability that a system, maybe only a single particle, propagates from state $x_i$ to $x_f$ in time $t_f-t_i$. Probably it is more difficult to understand that this notion is still maintained when the path integral is used to calculate the grand partition sum in quantum statistics.
Is there any understanding to be gained by learning how the technique was originally developed?
The idea of symbolizing formulae by nodes and connections between them is used in many other fields and was probably not new at the time. The idea is basically that of an isomorphism between a class of graphs and, given an unambiguous translation rule, the formulae at hand. This gives intuitive connection to graph theory and eases its application, for instance when a diagram is called 'connected' or 'disconnected', meaning that the respective formula can be factorized or not.
Another example of this kind that is not related to Feynman is the diagrammatic treatment of the classical Ising model.
Yes, there is such a thing, and Feynman diagrams were not invented by Feynman, but by mathematicians in the 19-th century, starting with Arthur Cayley. See, for example, my MO answer:
https://mathoverflow.net/questions/168888/who-invented-diagrammatic-algebra/260016#260016
A Feynman diagram is just an extremely compact yet precise notation for a contraction of tensors, say something like
$$
G_{i,r}=\sum_{j,\ell, p,q=1}^N C_{i,j}V_{j,\ell, p, q}C_{\ell, p}C_{q ,r}\ .
$$
If $C$ is a lattice propagator in position space and $V_{j,\ell,p,q}=\delta_{j,\ell}\delta_{j,p}\delta_{j,q}$, the above is just the tadpole diagram in the two-point function of the $\phi^4$ QFT. In the physics context, one wants to take $N\rightarrow\infty$ so summation over indices become integrals over continuous variables of position or momentum.
One can certainly do computation entirely in terms of diagrams, especially in the context of representation theory and invariant theory. This is the spirit of the book by Cvitanović, for instance.
You can also see examples of diagrammatic computations related to Pascal's Theorem in my article with Chipalkatti
"On the Reconstruction Problem for Pascal Lines",
Discrete & Computational Geometry volume 60, pages 381–405 (2018). Preprint version is here.
Best Answer
Our physics department had draughtsmen who drew them on "vellum" with Rotring drawing pens or something similar, and templates. The author produced a sketch and the professional made them publishable.
The draughting office charged a fee to your research grant, so drawing software saved us money, but at the same time made the draughtsman unemployed.