How can I maintain PWM output while still performing other tasks? I want to generate a PWM signal that I can turn on and off with a button press without using interrupts.  Is there a way to keep the PWM output going while running other processes?
PWM just runs once started. It's all managed by a completely separate bit of silicon inside the chip.

Once you have done analogWrite(...) the PWM will just keep going until you tell it otherwise.