Why do I need transistors?
Because the shift registers can't delver the current you need at the voltage you need.
I couldn't find any 2N2222 transistors, only 2N2222A ones. Would these work as well
Yes.
Between the registers and the transistors are resistors.
Yes the wiring sounds right.
Does it matter how many LEDs I have in each string?
The more LEDs the more current you need. Make sure this is less than 80% of the current rating of your transistor.