Eliminate back feeded Audio Signal
It's the same problem facing the simple telephone in a two wire connection - how to stop the amplified microphone signal being picked up by the earpiece receiver and getting too much local signal in your ear (side tone).


  How can I prevent this?


So, the sidetone cancellation circuit relies on a wheatstone bridge and controlled line impedances like so: -



TX or transmitter (amplified microphone) drives onto the line via a potential divider formed by the top-right resistor and the impedance presented by the line. This means that the RX receiver circuit has equal balanced signals from the transmit amplifier potential divider and the line and therefore receives NO TX signals.


  My Idea was, to use a transistor to get the original Signal and the
  inverted Signal and add those together to get (almost) no feedback.
  
  The other idea is to use an Inverting Opamp to Shift the Signal by
  180° in phase and add the Signals together...


You can try all sorts of things and what you will ALWAYS find is that you need controlled line impedances and a wheatstone bridge to get cancellation of TX and good reception of RX. Here is another more practical implementation: -



It still uses a wheatstone bridge and this relies on controlled line impedances. If you can come up with a linear technique that doesn't use line impedances then I will worship you as a god. Emphasis on linear meaning that a method that makes judgements based on amplitude and throws a switch to control audio direction doesn't count as linear.

Here's how the original old fashioned telephones implemented a wheatstone bridge using a transformer and carbon granule microphone: -



It's quite sophisticated in its operation with a reliance on the right turns ratios BUT it still needs the correct line impedance to cancel locally generated audio.