That is to say, what is my ruler that is untethered to the surface of the balloon?
Relativity probably doesn’t work this way at all, but if light did actually slow down, would it be perceived by us as light moving at the same speed across a greater distance?
As to the previous question about relative size. If the universe doubled in size, the objects within it such as atoms don’t also double in size. Gravity and the atomic forces don’t change, so eg the size of stable electron orbits or planetary orbits don’t double in size.
It is not the meter that is growing, that's impossible. Instead you must see it as it is, which is that there are more meters continuously appearing in all of space.
Do you mean all particles are moving away from each other? I think the question here was what would it mean for "points" to move? What are points anyway?
It is easier to understand in 2D: Imagine a deflated balloon with two dots drawn on it. When you inflate the balloon, its 2D surface increases, which results in the two points "moving away" from each other, without either of them actually moving. It‘s just that the surface on which they lie grows. This is also the reason why distant galaxies are getting more distant faster than the speed of light: They are not actually moving, but the space in between us and them grows.
Notice that the farther two points on the surface of the balloon are apart, the faster they will appear to be moving away from each other, which is why the effect is unnoticeable over small (in astronomical terms) distances. Gravity pulls things together much faster than space expansion pulls them apart.
Bringing this back to your example: Imagine "blowing up" the box. Things inside it are growing a little bit (but gravity pulls them back together), but everything seems to be moving away from everything else.