And the result is... not that good
What NeilM said:
First the slop in the bottom bracket: The bearings are loose or worn, so the BB need regreasing and adjusting, or replacing.
Second, the chainwheels are not square on the taper. Watching the video, if you look at the front of the bike, rather than where the inner chainwheel is touching the chainstay, you can see that either the spider is slightly bent, or that is is not running square.
Either way, I think you need to strip the BB for a rebuild and at the same time check the condition of the tapers on the driveside crank arm.
First part is correct: after cleaning, putting new grease and reinstallation the slop is completely gone and the axle runs smoothley. So that problem is fixed. But, the chainweel still keeps touching the chainstay on every rotation. I checked the condition of the taper and that one looks fine to me. I also cleaned the inner part of the crankarm so it looks strange to me that the problem is situated here.
What I'm gonna do: try to fit another DS crankarm and see if the problem is still there.
If so, I know it's the taper and I'll need a new axle (can you just rebuy those? Will they fit the cups, cause they have the french thread and are difficult to find/expensive?
If the problem is gone, I know it's the spider of the crankarm and I'll need a new crankset i suppose (I guess it's not fixable with a vice and a hammer