The LED's have arrived after the impulse decision to finally go through with the project, they always end up being a lot smaller than I think they will, even knowing the dimensions beforehand. I did manage to solder one of the LED's to some copper wire, so the project is off to a good start. Mind you, I did melt one in my first attempt, lucky I have 20 or so spare.
The LEDs are quite bright, I ended up using a 20-ohm resistor. With a 5V source, this gives a forward current around 100mA (80mA when checked with a multimeter). This is around 60% of the rated current for the LEDs, Even with this relatively low current, it's still going to require some more complex driving circuits than just some shift registers to light them all up.