[Scott Douglas]

My observations from out here in CA are these.
Your yard will expand if you remove it.
Your yard will also change character as going to a white fence from what you have will be quite a change IMO.
You could easily get rid of it saying 'You take it down, you haul it away, it's yours.'
I would consider cleaning up the in-between part before you take it down so you don't have to live with looking at it until it's done. That way you could do parts at a time and not have an all at once type thing going on.
Taking it down and cleaning up the boards (removing old nails etc) for use later is going to take some effort. Who puts in that effort will/should be reflected in the price you get for it.