It is finally working.

Built the whole thing, assembled, then redesigned most of the pen holder.

The problem was the following:

• The SG90 microservo I intended to use was not strong enough
• The line I wanted to use to pull up the Z carriage was a bad design decision
• The arduino was not able to provide enough current to the stalled microservo, so it rebooted when the pen pulled up.

In the new design:

• The SG90 replaced with a more powerful MG995
• The servo moving the carriage directly
• Added a 6V regulator for the servo, what gets its power from the beefy 12V PSU and not the arduino.

The electronics on the head still messy as I don't designed a proper PCB yet (but it will only happen when I finish the milling and the laser engraving head design):

And finally here is the first video of the working unit: