How do I manipulate a large number of moving platforms?
You could actually do this relatively cheaply using a mechanical solution, assuming you're happy to put up with a fairly long reconfiguration time. What you need is an X/Y axis CNC setup, a servo, a whole bunch of worm gears, and a small motor.

The idea is that you'd build each platform with a rack on the bottom, connected to a worm gear, such that the platform would raise and lower when the worm gear turns. The reason for using a worm gear is that the platform will stay put even if no tension is placed on the worm gear. A rod connects to the worm gear, and has a small cog on the end of it.

From there, you mount the X/Y axis CNC bed under the array of platforms. On the "head" (I don't know the proper term) you mount a servo, and on that you mount a small motor with a cog on it. The servo rotates the motor chassis so that it sits clear of the rods I mentioned in the last paragraph. The X/Y axis moves to position, the servo moves the motor upright so that it now touches the cog on the rod, and the motor spins to adjust the platform height. The servo then moves the motor chassis back into the "clear" position, then moves to the next platform. Rinse and repeat for all platforms.

Here's a horrible MS Paint drawing:



You could reduce the complexity a little by omitting the servo and just maneuvering the motor around the hanging cogs, or by using an X/Y/Z axis CNC bed.

This ensures that your cost is pretty much limited to a CNC bed, the fabrication of the platforms (3D print them, maybe?) and a bunch of mechanical parts; the motor, servo, and controller stuff should be super cheap anyway.

You could even drive the whole thing from an Arduino stepper shield. With a bit of cable management you could trivially wire LEDs into the bottom of the platforms, or you could go for something fancy like leaf springs to connect to a pair of plates on the side of each platform.