First, please tell me what exactly are these noises? In the 2nd image, they don't look like harmonics:
If they are harmonics. They should be resultant in the waveform. But my audacity waveforms don't show any resultant. The noises are riding in each part of the sine wave. How do you differentiate between EMI, RFI, amplifier noise, and quantization ADC noises? How sure are you they are powerline noises or harmonics? Please tell me what exactly those are first.
You seem to be looking for an individiual frequency as the source of your noise/distortion. It actually appears that this is broadband noise/noise floor that you are seeing. Also, a hint when using the audacity spectrum viewer - you need to amplify the signal first, as the viewer has a fixed minimum on the Y axis. You can then magnify as required in the spectrum viewer.
This is the spectrum of the last waveform you posted. I've also calculated normalised amplitudes for narrow band noise, on the basis that the signal is 10 uV rms.
You can see the 50 Hz signal, and small 2nd and 3rd harmonics. You can also see a small amount of 60 Hz noise (1 uV), but no harmonics.
However, there are 2 unknown and more important noise sources - 78 Hz (1.7 uV) as well as harmonics of 78 Hz, and broadband noise (approx 100 nV/sqrt Hz).
The 60 Hz is likely coming from power line noise, and could be mitigated with better shielding, or the use of a notch filter (depending on the frequencies of interest in your signal), or differential measurement.
The 78 Hz is a mystery, and may be noise generated by the simulator.
The broadband noise may be the noise floor of your setup, and because of it's 1 kHz bandwidth, it has a total contribution of approx 3 uV.
I suggest repeating the measurements with other waveforms and amplitudes, to see what happens to the unknown noise sources. It may be that some of the unknown noise is from your waveform generator.