Have now tested without the tach wire. No difference.
Do you mean the centre carbon pick-up inside the cap? Everyting look to be ok with it.
The spring contact on the rotor shows a small mark after contact with the carbon pick up, like I suppose it should.
As for grounding: 3 wires are connecting the cap with the coil and the rest of the assembly. This is like it was from the factory. I have not done anything to these. One of them are black and it is marked "ground".
Here are some more photos. If you see anything wrong, please let me know!
Cap installed with the tach and batt wire in place: