cosmology - Were fermions ever massless?

In a discussion of the Standard model and Higgs mechanism it was claimed that accordingly:

"During an early phase of the cosmos all fermions were massless."

I wonder whether this claim can be challenged. Therefore the following more detailed questions:

1. 
   

   Is it fair to say that, according to the standard model, the mass of any fermion is an "effective mass", that it may vary depending on the Higgs field strength, and that it must necessarily vanish only if the Higgs field strength is exactly zero?

   
   


2. 
   
   

   Is it fair to say that under conditions "during an early phase of the cosmos" (or: at sufficiently large temperature, density, ...), and while fermions are present, the Higgs field strength is not necessarily exactly zero?