I am very interested about your conclusions regarding the rotative magnetic field created by a 30 mm neodymyum sphere fixed between the poles ( 3500 - 34.000 RPM) first without the coil PS: I know 100 % how to use multimeteres since 20 years. I never asked about this. so far I did something wrong?
Until I wind another coil I can only tell you what I found by trying out your first video. The magnet being round is terrible at producing magnetic fields inside a coil, if you take a bit of paper and put on top of a square magnet, then sprinkle iron powder on the paper, you will see clearly defined magnetic fields. Do the same with a sphere magnet and the fields are not as defined.
Also a round magnet gives a really dirty scope output, noise and spikes all over the place. I will find you a video of a guy who built a wind turbine generator, in it he tells you why a square magnet that just covers each coil works best.
Your mistake with the multi meter was the setting and and using the Amps socket. Your meter has a milli Amp and a Amp socket, if you plug the lead of the meter into the mA socket and switch the range over from Amps to mA on your meter you should be able to read down to mA. So instead of getting 0.00 your meter should should show .000, so you should be able to see something like .001 or whatever.
From that you can work out your actual Watts produced by the coil, that way when you mes with different magnets and move the motor away but keep the magnet where it is you can take proper readings. in this case its border line you will light the leds without the motor being close.
The other thing I think is going on, is the magnet is acting like a core in a cored inductor, again I need to wait until my magnet wire arrives so I can give you the numbers. Once you can take real readings you have a start point, this will at least let you know how much each thing you change affects things.
69% efficient is a number you based on the assumption that a 7W led light uses 7W of power from the coil. In this case it dosnt! It is nowhere near it. Get a torch light bulb (not led), something rated for say 3V or 1.5V and replace where the led is, I bet it wont glow. Even at your 240V its unlikely you will light the 1.5V bulb.
But again I cant try it until the ebay fairies arrive with my new wire.