The resistor will depend on the LED.
Red, generally are 2V.
Green, Blue, generally 3.3V.
The Transistor will have voltage drop from collector to emitter, ~0.7V
So do some math:
( (Vsource) - (Vce-transistor) - (Vf-led) ) / (current desired) = Resistance needed
(5V - 0.7 - 2)/0.02 = 115 ohm
(5V - 0.7 - 3.3)/0.02 = 50 ohm
Check the parts you want to use and put int the right values.
For the splitter cable - hard to draw graphically. I can't open your attachments.
The arduino will draw what it needs, like 40-50mA.
The 30 LEDs will draw whatever you decide to let them draw via resistor selection.
30 LEDs, 20mA, so 600mA.
So use that to size your supply requirements.
The barrel jack connector output goes to a reverse polarity protection diode which then connects to the Vin header pin, and then feeds the 5V regulator and shows up at the 5V header pin.
If you want to use 7.5V to supply the barrel jack, or the Vin pin, and the LEDs in parallel (adjusting your resistor values accordingly) that would be ideal.