Looking at this orchestral performance of sinfonia from Bach's 29th Cantata (BWV 29), it is really curious to see the trombonists (?) holding the horn/trombone with one hand and the way they are standing with the other hand on their hip. You can see it at the start of the clip and then throughout the sinfonia (the first three and a half minutes).

First question, what instrument is that? Is it a trombone? Some type of a horn?

Second, why are they holding it like that? All three of them are doing it throughout the whole performance. Seems a little strange. Is it a stylistic thing? Is that how people held that instrument four hundred years ago?