Does the square-wave nature of PWM dictate how much resistance you need?
It's... mostly wrong. For three independent reasons.

The LEDs themselves, like all diodes, absorb some of the electrovoltaic potential of the electricity passing through them, resulting in a voltage decrease in their circuit. This means that although 5V is being used to power the resistor and LED in a circuit, only some of that voltage is passing through the resistor. If we use a typical (for blue, white, and other exotic colors) forward voltage (Vf) of 2.8V, this means that the resistor only has 2.2V across it. For a 200ohm resistor, this means that only 11mA is passing through the circuit. If you want 20mA to pass through the LED you need to use a resistance of 110ohm instead.

Second, the voltage from the MCU pins themselves depend on the current passing through them. Section 32.7 of the ATmega2560 datasheet, "Typical Characteristics - Pin Driver Strength" has charts for voltage versus current for both sourcing and sinking current; at 20mA the voltage drops to about 4.5V, which means that the resistance required for a 2.8V LED drops from 110ohm down to 85ohm.

Third, although semiconductors (i.e. the MCU and LED) can be damaged when too much current passes through them, it is not the current itself that causes the damage. The current passing through the voltage drop results in a consumption of power, and this power subsequently turns into heat. This heat then causes the breakdown of the semiconductor material. Even a junky, third-hand LED can take bursts of current up to 35mA or so without being damaged, provided the duration is limited. With a PWM signal this duration is directly linked to both the frequency and the duty cycle of the signal. You will need to see the datasheet for the LED in question to figure out what the maximum safe current and duration are, but the implication is that if you keep the frequency high enough and the duty cycle low enough, you can get away without any resistor at all and both the MCU and LED will survive.