Thanks for your responses.
Based on your feedback, I've done the calculations below. I've also never done this sort of thing before so if you think I've got something wrong please let me know.
I am going to build this circuit with blue LEDS (if the parts ever arrive - geeze it's taking forever).
Regarding the 220 ohm resistor to protect the LEDs...I have a data sheet for super-bright blue LEDs that says the forward voltage is 3.5 volts. The 'internet' also says the forward voltage for blue LEDs is 3.3 volts. So assuming the lower value, each LED would draw 7.7 mA (i.e., (5-3.3)/220).
The shift register current limit (from a Philips datasheet) is indeed 70 mA as stated above. At most, the shift register would be supplying 8 LEDs simultaneously, which means the maximum shift register current is 62 mA (8*7.7) - so I think 220 ohms is ok for blue LEDs.
For the transistors, they may be ok as is due to the short ON time but to be on the safe side I will add a resistor for protection. At most, each pair of transistors will be powering 64 LEDs, which means the collector current will be about 500 mA (64*7.7), or 250 mA per transistor (he has two for each layer of the cube but I think he could have managed with one because the 2222a version can take 800 mA). Assuming 1/10 of the collector current to put the transistor into saturation means the base current should be about 25 ma, which means (assuming a 0.7 base emitter voltage drop), a resistor of 172 ohms should do.
I have some 110 ohms so I'm thinking those will be ok, since that will ensure more than 25 ma but not exceed the max rating from the datasheet of 50 mA base current (for saturation at 500 mA collector current).
Does this seem reasonable to you?
I'm also thinking a decoupling capacitor should be provided for each shift register, also based on what I've been reading on the internet. As I understand it, this is to protect the shift register from inductance induced voltages which could go high temporarily as the shift register changes state. Presumably you would also agree with that?
Lots of changes to this circuit, and I'm going to take the button out as well and add more effects to the software. I hope it all comes together - a month ago I had no clue about transistors, shift registers, diodes, microprocessors and C programming. But I did know Ohm's Law