Will 74HC595 as LED sink damage LED when pin goes high?
The answer lies in the reverse voltage rating of the LED chosen. Common through-hole LEDs typically are rated at 5 Volts reverse, some LEDs higher. This means that for up to 5 Volts applied in reverse bias, the LED will behave as a regular diode, blocking current flow. Beyond this voltage, reverse breakdown might occur, potentially destroying the LED if the voltage is high enough. 

In the schematic shown, the PWM signal is 0 to 5 Volts. Thus, even when the PWM signal is low, and the latch output is high, the maximum reverse voltage the LED is exposed to is less than 5 Volts. So the LEDs will be fine.