Electronic – Problem controlling common cathode 12V RGB LED

As long as you've got some transistors to drive the LEDs, then you can source your 16-bit PWM from a PSoC3. A single PSoC microcontroller should be able to give you 8x3 16-bit PWM channels. You'll then have to multiplex them.

The PSoC should even be able to handle the multiplexing in hardware, using DMA to stream the image data from memory, and deliver it to the PWM channels. This leaves the CPU to do other important tasks like generating the display data

The PSoC5 has an ARM Cortex M3 core. The good thing about the PSoC chips, though, is that they have re-configurable digital peripherals. This means you can choose to have loads of PWM, or loads of SPI, UARTS, or any combinations. You can wire up all of these peripherals internally, with gates, timers, flip flops, etc.

This means that you can probably make the hardware in the PSoC manage the screen without any intervention by the CPU, giving a good solid image.

Yes, you can make an 8x8x8 RGB LED cube with common-cathode LEDs.

Unfortunately, it won't be any simpler than a similar cube made with common-anode LEDs.

The reason is simple: it can seen from symmetry that common-cathode isn't any different from common-anode except for swapping the cathode and the anode:

^{simulate this circuit – Schematic created using CircuitLab}

The driving circuitry is just the same, except you swap positive voltages for negative voltages. Instead of NPN transistors, you have PNP transistors. Instead of current sources, you have current sinks. Just different. Not simpler.

You could design an LED cube with fewer shift registers regardless of the choice of common-cathode or common-anode. However, you then have to multiplex more LEDs, meaning each gets a smaller slice of time in which it can be on. At some point, the duty cycle of your LEDs becomes so low that you can't reasonably make them bright enough.