Does transformer's flux (due to load) need to obey Faraday's law
Ignoring magnetization flux
The magnetic flux produced by current entering a transformer's primary (due to the loading on the secondary) is perfectly cancelled by the magnetic flux produced by current leaving the transformer secondary: -

Picture from here.
Taking into account magnetization flux

The core can never saturate when supplied with rated voltage

In the real world the saturation of a core always happens but it's usually a &quot;reasonable&quot; amount that doesn't cause excessive heating.