Hi again, I had to leave this proyect on standby a bit, due to work.
It's very clear to me now that it's a matter of power, mAh...
I hooked up the mp3 module to the arduno board, with only a small sound sketch that I got from the DFPlayer library, on the computer through its USB port, and worked like a charm.
As soon as I hooked it up on my powerbank, nothing happens.
Finally I got a datasheet of the powerbanks and they run only 2600mAh, with 1A, 5V.
So far I have found out that:
The arduino itself already uses around 500mAh.
The standby function of the mp3 player uses 20mAH (I have no idea what it uses to play sound, haven't found that data yet).
Over the computer's USB, the voltage meter shows 5,02V - 4,98V and it works
Over the powerbank, the voltage meter shows 4,87V, the led on he MP3 player doesn't light up (it should when the sound starts to play) and nothing happens, not even a change in the voltage meter.
The speaker is a 5V, 3W, 8Ohm little speaker (no idea what the mAh usage is) but AND I MAY BE WRONG, if the speaker is 3W, with 5V, the amperage should be 600mAh (5V x 600mAh = 3W).
So :
500mAh for the arduino + 20mAh for the MP3 + 600mAh for the speaker = 1120 mAh, which is too much for the powerbank to handle if not fully loaded (I think).
The USB Port of the computer will always give a steady V, mAh or W.
Am I right?
Completely wrong?
Does anyone have a sketch that does sound and lights at the "same" time or even one after another?
Now I am really at the end of trying to find out what the heck happens ... 
Hope someone has an answer