Problem understanding when the OPAMP is switching
In case that NO energy is in my system, does it oscillate? -> Falstad shows an oscillation without any source connected (maybe
  noise?)
  


With no power source, it won't oscillate. Oscillators need a power source. Falstad is probably applying generic supply voltages to the op-amp


  
  Since Uc is decr/increasing to U+, uC never reaches U+, so WHY does
  the Comparator-Output Change?
  


This circuit will oscillate if the power rails (secretly applied by Falstad) are bipolar (say) + and - 15 volts. This puts "0 volts" at the mid rail of the supplies and the capacitor can reach the upper and lower thresholds as dictated by the resistors in the positive feedback loop.