Normally, the LED strip comes individually with a capacitor and a resistor for each LED chip, in this case for the cube I would like to make only the LED chip apparent.
My question would be:
Can I use a capacitor under each 8 led tower? If yes what would be its value? My intention is to keep the capacitors hidden under the platform.
Do I need a resistor in each LED chip or can I use only one resistor in the data input of the first LED? If yes what would be its value?
Remembering that in total I will be using 512 LED chips.
No, it must be one cap per led and these must be very close to the led pins. They are called "bypass" caps and prevent the chips in the leds from causing malfunctions in the chips in the other leds. So for the caps to work best, they should be the closer to the chip than any other component.
Are you certain there are resistors on the strip? Normally there are only capacitors.
The same as the resistors you are advising at the start of the chain. In series with "Din".
As these are individual units intended to be connected (and can be purchased) with length of ribbon cable between, it makes sense that the resistor is included to suppress any transmission line effects in between. The resistor of course has no deleterious effects on the signal between LEDs.
You can also obtain the single chip modules without the resistor - but with the capacitor.
Of course you would need to add a capacitor to each if you used the 5 mm axial version of the LEDs, it should be possible (just) to solder a chip across the supply leads.