How to interpret the FFT of the mains voltage in this case?
I marked the harmonics of the fundamental 50Hz in red color. As you
  see the highest harmonics are at 150Hz, 350Hz, 550Hz, 750Hz. These are
  the 3rd, 7th, 11th and 15th harmonics. Is that order of significant
  harmonics something common in the world or completely random?


They are common and are generated from transformers and other components


  Complex waveforms are generated by common electrical devices such as
  iron-cored inductors, switching transformers, electronic ballasts in
  fluorescent lights and other such heavily inductive loads as well as
  the output voltage and current waveforms of AC alternators, generators
  and other such electrical machines. The result is that the current
  waveform may not be sinusoidal even though the voltage waveform is
  Source: https://www.electronics-tutorials.ws/accircuits/harmonics.html  


.


  I marked the highest non-harmonic FFT components after 50Hz in green.
  They are 60Hz, 70Hz, 80Hz so on. I'm confused here because I thought
  the real distortion in mains would come from the harmonics. But here
  there are 60Hz, 70Hz, 80Hz ect. whose amplitudes' are greater than the
  harmonics. What could these be? Am I interpreting something wrong
  here?


No, they are not really peaks, just part of the 50Hz signal with noise. In an ideal world, with perfect sine waves if we took the FFT it would look like a spike (or dirac delta function if you want to get technical). Because the real world is noisy, with phase and amplitude noise, when we observe noisy sine waves. The difference looks like this and is also common:


Source: http://www.dilettantesdictionary.org/index.php?search=1&amp;searchtxt=modulation