currently I am working with ArcGIS 10.4. My project contains several layers such as streets and buildings, as well as retail and residential areas for a large-area state in Germany. As this is a lot of data and I am currently focusing on a smaller region within this state, I am looking for an easy way to “cut” all data outside my region of interest, but retaining the symbols and the labels. To sum it up: Is there a way to do something like zooming into my project and cutting everything else outside in one step?
You can use Clip tool under Analysis menu. (You can access Analysis Menu by clicking red toolbox on toolbar). Clip will help you clip the dataset by using a clip feature (in your case it's a polygon of your area of interest).
Since you want to clip multiple layers you can use Batch Clip by right clicking on the Clip tool.
Explanation regarding clip tool: http://desktop.arcgis.com/en/arcmap/latest/tools/analysis-toolbox/clip.htm
It does not seem clear whether you want to actually or just graphically clip those multiple layers.
Consequently, I think Clipping the Data Frame may meet your requirement:
If you need the data inside a data frame to be a shape other than rectangular, you can clip your data frame's drawing to meet your map specifications using the Clip options on the Data Frame tab of the Data Frame Properties dialog box. Clipping does not alter the actual data inside the data frame, even though data view also shows you the clipped data. Labels honor the clip shape. They will redraw so you will not see any clipped labels.