>How would you describe in a single sentence the growing precision of our calculation of the speed of light constant over the last 300 years?
I don't know, first I'd talk about how it was discovered that light behaves like c really is a constant (that's not intuitive nor trivial). You could guess from Fizeau (1851) and Michelson-Morley (1887) experiments, but it wasn't really until the 1905 Einstein's paper that it was understood that you don't need aether, the speed of light can (and should) be a universal constant and it all makes sense.
>How would you describe in a single sentence the way Einstein used Mercury's retrograde motion to assure himself of the advocacy of his theoretical mathematical model for gravitation?
He shows (see here: http://www.gsjournal.net/old/eeuro/vankov.pdf) that his theory (after reasonable approximations) predicts a precession of the perihelion of Mercury that agrees with the experimental result, then he stops. That's what you do in Physics: look, my nice theory predicts this observable in agreement with the experimental data. That's a very good sign, but the data could be explained by other theory as well. Data can "prove" that a theory is wrong (if your theory has falsifiability, as it should), but they can never show that your theory is all that there is, nor your maths... There are other theories of gravitation, for instance Logunov's, that are interesting and agree with data (more or less) but have a few problems with maths.
BTW, I don't think he ever really doubted his GR theory, it comes from a very simple, yet profound idea that once realised had to be right. You have to do all the technical work afterwards, that was difficult, but a few mathematicians could have done/did it (Hilbert). In any case the idea is to check your predictions with the measurements from more and more precise experiments to find out if you're wrong, not the other way around (that... well, I've met that funny epistemology a few times in social sciences.)