So far I have stumbled upon two problems:

- The ESP-01 did not produced a stable signal to drive the neopixels either using the Adafruit library or the library that uses the serial port.  That is the reason why I have added  the Arduino board.

- The Data In lane of the neopixel strip that I have bought is certainly on the component side. I haven' t notice the importance of the via on the strip until It was too late an I had cut all the pixels apart. On the image below the vias can be clearly seen.

As I have soldered all my LEDs from back It was necessary to change the connection on 4 of them to the other side. As none of them presented problem in the DOUT lane I suppose that in my strip the DOUT was on the solder side (or both).