the DAV was formed in 1938 after the anschluss in Austria from former members of the Deutscher/Austrian Mountain Verein that was formed around 1873.
It was disbanded by the Allies at the end of WW2 so that dates your badge to the period 1938-45 IMO.
Both the badge and the box look like period originals to me.
All the best
However: The German Alpine Club was re-established in 1952. It joined the Deutscher Sportbund organization in 1992. After leaving the Union Internationale des Associations d'Alpinisme in 2008, mainly due to differences in respecting both competitions and recreational sports, the German and Austrian Alpine Clubs rejoined the Union Internationale des Associations d'Alpinisme in 2013.