Help check my (basic) understanding

So, I've gone back to a 4x4x4 and am having a conversation with myself regarding the need for transistors. As the Uno can utilize 20 pins (16 columns, 4 rows) as digital pins and because of multiplexing, shouldn't I be able to (safely) get away without transistors?

No. Think about a given pattern you may want to display that turns on all 16 leds of a single level (Row) going to a single output pin driving that row. That could be 16 X 20 milliamos of current which is more then the poor row enable pin can supply, hence the need to use 4 switching transistors to drive the 4 levels (Row). That make sense?

Regardless, I am amidst some heady (for me) reading on transistors and hope to understand the use of them for this project if necessary. I have some 2N2222's here and was initially planning on using these on the rows but am having some serious trouble understanding the calculations behind choosing correct resistors to use with the transistors. Any "beginner language" help would be greatly appreciated.

A 2N2222A transistor is really not able to switch 16 x 20ma of current, you need a 'bigger' transistor.

I've also attempted a first draft of a schematic for this project (attached). It assumes the use of transistors but does not include the connections to the Uno. I plan to have the columns connect to "digital" pins 0-13 and "analog" 0-1, while the rows will connect to "analog" 2-5. Thoughts? Am I tracking in the right direction?

Keep in mind that pins 0 and 1 are used to upload sketches into your chip, so you may need to be able to easily remove things you wire to pins 0 and 1 when you need to upload a sketch and then reattach them.

Nickity - I am planning on "diffusing" the LEDs by sanding them. While this won't diminish the brightness, I hope it will prevent the need for sunglasses.

Others have tried many methods of manually diffusing clear leds with rather poor effect, including sanding, painting, etching, etc. But maybe you will have better luck. One thing is clear that led cubes should definitely utilize diffused leds, otherwise the visual effects lose a lot if the leds have narrower output beam, making the angle of viewing of the cube poor, the effects just won't be as good as using diffused leds.
Lefty