Arduino Uno Issue: No output power
The schematic you have drawn is completely safe to connect to an Arduino PWM pin. It may not be an optimal choice (closing multiple switches will change the brightness of segments), but it's not dangerous. As you said yourself, the current is limited to a safe value even if there's a dead short in the 7-segment display.

Typically, you burn your board when you connect/disconnect stuff while your Arduino is powered (because intermediate circuits may be unsafe even if the final result is OK), or when you built a circuit which is different from what you think you have built.