Why can't I plug the 12V 3A PSU directly into the Arduino then run that power out the Vin plug?
Because the diode between the power jack and Vin is only rated at 1A.
Why are there resistors before each transistor
They are not transistors they are FETs. They protect the arduino from too much current.
I would round to the second number because you always want to play it safe and go to the closest value above the calculation right?
Yes
I don't see how to wire the anodes for the LEDs.
To the +ve of the power supply.