DFPlayer - A Mini MP3 Player (DFR0299)

This is causing the loop.
ADKEY1 & ADKEY2 are analog inputs, changing resistor changes function. See Wiki Doc.
Changing the jumper to a momentary switch will stop it looping.
You could try the pause/play (33k) as per diagram below and see it that causes a loop.

Playing a file on power down depends on how you power it down.
Simply switching off power will not allow time for a file to play, you would need to delay the power off. This could be done by the way you connect the switch and other components.
You could use a mosfet to provide power and a capacitor to hold it on long enough for the file to play when activating the power off switch.

Regards
Jim