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I started with a simple case, does anyone know how this cluster of planets will move if starting from stationary?

it's really hard to see why this would not start collapsing towards the center. After all there is no initial kinetic energy stored in the system to prevent that.

But if it's an equilibrium, as you demonstrated by calculation - it would be an extremely unstable one, right? One butterfly flaps its wing and the thing comes crashing down.

ehrm, sorry, but the last 2 pictures would collapse !
directly to the center!

Because to the left is a 0.5kg planet 1m away (so has a force of 0.5/1²) and left of that is a 0.25kg planet 1.5m away.
I didn't write the number of kg on all planets, I hope you can see that it is an infinite series that doubles in mass and doubles in distance.

The strength of gravity decreases with the square of its distance away.

As I drew on the last picture, the planet has the pull of large masses to the left (but quite far away) and small masses to the right (but very close), so the idea is that mass/dist² is equal on both sides of every planet... since none of them have a resulting force, none will have an acceleration, so won't move.

On second thoughts, it does seem like its a trick. A bit like a levitating turtle because he's sitting on a smaller turtle which is in turn sitting on a smaller turtle etc.
So I wonder what the motion really is, or if it can really be calculated given the infinite number of bodies.

Interesting as usual Tom.
This is a case of unstable dynamic equilibrium. I think Steven Hawkings did the calculations with his students for his programme Steven Hawkings universe.

Do you know of the "paradox" of the athlete and the tortoise? The tortoise moves exactly one tenth of the distance the athlete does. Can he ever pass the tortoise if he starts 10 stadia behind him?

Well, yes in real life we regularly beat tortoises in races! The lesson here is to free your mind from whatever constraints imposed on it and think outside the box!

At the "time" this question was posed Europeans had no concept of time as we do today. In fact it was quite common to denote what we call time by a distance. This is principally what Gallileo was attempting to establish in science: a  scientific convention of time. The period of a pendulum was eventually adopted for this purpose and Huygens really did become the father of scientific time!

Can we please give the poor butterfly a break? He didn't do it! It was a cilia on the outside of a bacteria!!