Everything you mentioned, including streaming Video,
is definitely possible on the mini.
As long as the streaming is not Flash then you are good to go.
YouTube has converted most of their content from Flash to HTML5 which give a much better quality picture.
Apps like HBO to GO and DIrectv both have a very good streaming system for movies etc. You do have to be an HBO subscriber at the house though.
Music and podcasts are easily handled in iTunes, as is some video streaming.