Which of Perutz et al's protein structures other than 3₁₀ helix were correct?
I think you've got the list of good predictions from the Bragg, Perutz, and Kendrick paper.  And the 310 helix was not really right either - it did turn out to show up occasionally in protein structures though. 

At the time all of these secondary structure elements were well evidenced from noncrystalline diffraction data and small molecule crystal structures. Examples being collagen from hair and spider silk fiber diffraction patterns. Fiber diffraction of hair, whose ordered structure consists mainly of alpha helices somewhat aligned to the fiber axis (along the length of the hair) or the mainly beta sheets which constitute silk.  

Because it was known that these substances contain mainly protein, building models which would show how regular peptide polymers would produce these patterns.  This happened in chemistry and physics departments worldwide.  The paper you cite is representative to the field.  

Pauling caused a splash in the spring of 1951 by publishing descriptions of the alpha helix and the beta sheet (both parallel and antiparallel).  The main advantage that Pauling had was that he had studied the structure of simple di-amino acid structures (notably glycyl-glycine) and understood that the peptide bond had some double bond like properties which prevented it from rotating freely.  As the discoverer of valence bond theory, which united quantum mechanics and chemistry and established the basis for at least most organic/biological chemical structures Pauling must have been practically alone in this insight at the time. 

This next set the stage for the dramatic race to discover the structure of DNA. If you read Watsons' memoir, he describes how the MRC in Cambridge was still feeling the heat,  competing with CalTech and Pauling.  He and the young lab there was holding their own against the MRC Cambridge with its multiple nobel laureates; a lab which had established the relationship of x-ray diffraction and chemical structure (the notes on chapter 7 are interesting - follow the link).  

Looking back on it, Pauling's main advantage was that he understood the peptide bond better than the rest of the world. By the time DNA was being resolved, the conformations of organic molecules were understood by many and the playing field should have been more level.  I'm not sure I buy the arguments that Pauling would have got the structure without seeing the diffraction patterns and even then maybe not. I might not give such strong credence to the idea that Pauling would have gotten the DNA structure so quickly if he had been able to travel.  

Of course Perutz and Kendrew verified the peptide alpha helix themselves when they solved the first crystal structures of hemoglobin and myoglobin, having the final say.  The structures contain a small amount of 310 helix but are otherwise full of alpha helices.  I'm not sure that the secondary structure was entirely confirmed until the 1958 when myoglobin was resolved.