Lets forget the wall business, the drainage issue is a biggie for me
Im guessing its a biggie to you, because its the only lever you against your neighbour.
Clearly its a lash up at the moment, but your neighbours have a downpipe not too far from the fence, it wouldnt be that awkward to replace the gutter with downpipe and route to connect with existing downpipe or go into a gully and connect below ground.
I hope you get the issues sorted soon......just avoid getting bogged down with PW surveyors