Electronic – arduino – Power 14 red LED’s and arduino from same source

Simply power the LED driver (i.e., its VCC pin) with a battery (or other higher-current-capable source), since the Arduino pins are only capable of sourcing 50 mA.

1. First of all, you won't even need a transistor, if you use a battery as I mentioned above. But to answer your question, if you were using a transistor: while there is lag, lag won't be significant enough to affect your application here.

2. And secondly, the TLC5940 is a constant-current driver, so you're fine. A constant current source/sink is exactly that -- it ensures a constant current flows independent of power supply fluctuations/other conditions. Thus your LEDs will light up with constant brightness.

Finally, a clarification regarding PWM with the TLC5940: You don't use PWM of the Arduino pin to directly create PWM on the TLC5940 I/O pins; that's NOT how it works. Instead, the TLC5940 uses serial communication with Arduino -- so you would send it instructions from the Arduino (via SPI or bit-banging, for example; see this TLC5940 Arduino library), and the TLC5940 reads those instructions and performs PWM on the desired I/O pins as instructed.

What you have guessed is correct -- if you want to run the LEDs in parallel, then you need to be able to source 600mA. If you run in series, then your power supply needs to supply at least 64V. I don't know the specifics for selecting the ideal power rail, but I would certainly shoot for >64V and put in the current limiting resistor.

A resistor on its own doesn't "drop 3.2V each" -- the voltage drop across a resistor is IxR, so the actual drop depends upon the current going through the circuit. You can calculator or measure the current going through your other group(s) of LEDs, and then pick the appropriate resistor for the mismatched group so that the current is the same.