Electronic – Reason for anvil and post in LEDs

I think you just answered your own question. If you want to power 25 LEDs @ 20mA that's 500mA total. So no you can't drive or sink 500mA with a single pin when that pin can only supply 40mA. Try using a transistor on the output that can handle the current. I think you're on the right track there.

Maybe try something like this. Disclaimer though I only thought about that circuit for a few minutes, I didn't try it or sim it. But that's half the fun :)

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.