Have you considered a vinyl fence yet? These dont have an unfinished side and of course never need painted and can't rot. Would look great as a pickett fence and just snaps together. Initial cost is a little higher, but I think its worth it.
As far as the mish-mash of fences around your property, it sounds like they probably belong to the neighbors. Roger's advice to inform the neighbors of your plans and address any concerns is very good. You are not obligated to do anything but comply with any CC&Rs and erect the fence on your own property, but its always nice if everyone agrees with what you are doing.