In regard to the hat questions:
Capello romano (my preferred term is saturno): probably not seen in the U.S. much before Vatican II, and is virtually nonexistent today except for your occasional eccentric trad priest. The saturno was much more common in southern Europe. It's strictly a secular clerical hat.
Biretta: It can
be worn everywhere, but it's a formal hat that's roughly analogous to the top hat in civilian fashion, I guess. It represents the cleric's teaching authority, and is directly related to the mortarboard/graduation cap. Both are descended from the medieval universities. As for wearing it non-liturgically, a priest could wear it while administering confession. He could also wear it while teaching a class. But today this might be seen as a little eccentric even among trad priests. The most realistic use of them, outside the liturgy, would be while the priest is walking on church property in his "house cassock".
Zucchetto: You didn't mention these, but since we're on the subject, don't forget that all clerics can wear them, not just bishops. These are the hats that resemble Jewish skullcaps. A priest's or deacon's zucchetto is plain black. This is even rarer than the saturno, but pictures of 19th century priests often depict them wearing it.
In conclusion, your priest character probably wouldn't wear any of those hats out on the street, outside church property. In general, American priests pre-Vatican II tried to keep a fairly inconspicuous appearance. In 1884, the Third Council of Baltimore
forbade clerics from wearing cassocks outside of church property.