What's wrong with: have you a TV?
Nothing, particularly in the UK. Anyway, DL accepts it (11/2015).
So is television Fernseher or Fernsehen? It seems be spelled both ways by Duo.
English "television" is ambiguous in that it can mean both a TV set (i.e., a device) and the medium (she works in television): http://tinyurl.com/l3sbr7e
I believe Fernseher is strictly a TV set while Fernsehen has both meanings.
Have you a television?