Transformer Harmonics
As mentioned in the link you posted, the voltage harmonics occur due to the voltage drop in series resistances of the circuit. The current harmonic components (caused by non-linear loads) generate these voltage drops. The voltage at the transformer terminals then becomes the original voltage added to these voltage drops (which contains harmonics), becoming a &quot;distorted signal&quot;, so to speak. About your thoughts:

The Transformer is a galvanic isolation, so no direct current can pass

Exactly.

Via induction and magnetic fields, effect is transmitted, i.e. also disturbances

Yes. Thinking of the periodic signal in terms of FT (Fourier Transform), each component of the signal is transmitted by voltage induction. For three-phase transformers with delta connection or arrangements of several connection groups (12 pulse transformers, for example) some harmonic components cancel each other and this is often used to reduce the harmonic content generated by non-linear loads (rectifying bridges, in general).

Turns ratio needs to be considered

Yes. The voltage induction for each component follows the transformation ratio.

The transformer will heat up due to harmonics if not it is a specially designed &quot;K-factor&quot; transformer. This affects the transmission behaviour?

Even a &quot;K-factor&quot; transformer will heat up, but this transformer is designed to withstand this heating without exceeding the maximum temperature allowed by the insulating materials. Part of the losses generated by the load current depend on the frequency squared, so components of high harmonic order have can greatly increase these losses, manufacturers usually take some action in the design to reduce that part of the losses that increase with frequency, for example, using conductors with smaller dimensions (of course this implies using more conductors in parallel). The voltage harmonics do not affect the transformer so strongly, in general they flow through the core, but their impact is inversely proportional to the square of the frequency.

Dependance on &quot;stiffness&quot; of HV net. If a power producer sits right next to the installation, it is hard to insert harmonics. Can this be described via the nets impedance?

That's right. The lower the series impedance of the grid, the lower will be the voltage drops caused by current harmonic components.
Edit----
Finally, answering the question: Each harmonic pass to the other side according with the transformer ratio. So, if you have 10% of 3rd harmonic in LV, you will have it in HV too. If 10% is significant, then you have a problem. In single phase transformers, there is no cancelling of harmonic components, it occurs only in three phase arrangements. The common practice is to filter it in LV side