First of all, thanks for your reply! To be honest, I think I solved my problem. But let me explain in detail, hopefully this helps others as well:

First, concerning the Question: "Why do you have a S-N curve with a large negative mean stress?":

If I have a part that is cyclically loaded in a zero-based manner, that means the load is applied and relieved periodically, but never inversed, some regions may see only compressive stresses (an no tensile stresses ever). A lot of material models assume higher fatigue loads for such compressive forces (which intuitively makes sense to me). And in order to consider that fact, I do need S-N-curves with negative mean stress values (since the compressively loaded area has a negative mean stress). Otherwise, the calculated safety factor is too low for such areas.

Concerning "The mean stress correction curves shown in the FT graphics window are just generic curves shown for visualization."

Good to know! Just a bit confusing that they show these (meaningless) curves...

Now concerning "There appears to be a bug when using a mean stress curve with a negative mean stress."

That was my first thought: A bug. But I guess the details are very crucial! In order to consider the S-N curve with negative mean stress, I have to assess the ABSOLUTE first principal stress (or absolute von-mises stresses). Because in the compression area, the first principal stress is not relevant but the third one. The one with compressive stress. By choosing ABSOLUTE first mean stress, I assume Ansys takes the first mean stress with the highest number, independent from the sign (+ or -). And then I get a safety factor of 1 for the compression side as well (in my test model described!

Now I hope I'm not just thinking I understood everything... 😬

Is it possible to upload an archive of my test model here? Would only be 228 kB...