As far as the flickering. Correct me if I am wrong, but we can wait to turn on the row transistors until the data string has gone through all the shift registers. That should keep all the LEDs from flickering.
Yes that is correct.
All I can can think of is that the ULN2003A chips have current clamps so they dont burn out the LEDs.
But they don't.
Do I have to design an alphabet?
Yes or use some one else's.
How do you handle the transistor rows when they aren't in sequential order?
You use a look array like it shows you in that link.