Electronic – Quick Question about ohms law and resistors

Your calculations are correct. I would give the resistor somewhat headroom, power-wise: if the LEDs' voltage is a bit less than 3.2 V, like 3.1 V (the voltage can vary by much more than that!) the current will rise to 22.5 mA, and the voltage drop across the resistor to 5.4 V. Then the power will be 120 mW, so the 100 mW won't do. Overmore, 1/4 W is a more standard value, and probably cheaper.

So go for a 240 Ω/0.25 W resistor.

This only answers one part of your question, but I think it might be the main place where you are misunderstanding.

How can the Ohm Law be applied in this case?

"Ohm's Law" isn't a law for all circuits. It's a description of one particular type of component: the ideal linear resistor.

It doesn't matter how your power supplies are arranged in your circuit. Ohm's law has nothing to do with power supplies.

The general laws you want to consider that describe how currents and voltages relate in any circuit are Kirchoff's Current Law and Kirchoff's Voltage Laws. These laws, along with specific current-voltage relationships for each type of device (one of which is Ohm's Law), are the main tools for analyzing circuits.