Trapp Family.
The “Von” was added for the movie, for some unknown reason.
No, he was a "Ritter" (Knight) and thus entitled to the 'von'. When the socialists took over Austria after World War I, they abolished the titles, but von Trapp was entitled to Italian citizenship, and Italy retained titles.
So, in Austria, the country that awarded the title, he would have been called Trapp; in Italy, whose ships he sank, he would have been called von Trapp.