15x15x15 LED cube with thermal sensors
The answer to "Can an Arduino even handle the processing power" is probably No, but no one can really answer because you have not specified the problem.

Just for starters you have >3k LEDS, if you want to control RGB that means ~10k devices.

To operate at moderate brightness (10mA per LED) you will need at least a 35A power supply. This will require external high power circuitry to switch/control.

Do you need to individually control each LED? To control 10k devices you will need serious external circuitry.

Do you need to control brightness or just switch individual devices on/off?

You could probably achieve a solution, but 15 modules each controlling 15*15 LEDS may be achievable, 32 or 64 is probably more realistic.

Either way soldering by hand would be impractical, you need to design a board for each module.

You could use an Arduino on each module, but a ATMega chip would achieve the same end, without the unnecessary peripheral circuitry.