Panamabob (Robert) has also replied (albeit very short, point form answers)
Count is performed on the crown (I'm not sure which bit)
For travelling and rolling, he suggests the 25 dollar white Cuenca.
"Any white cuenca will roll. Most natural hats over $150 will fold over nicely."
That's set. I'm getting the white 25 buck Cuenca in Cuencano.