Does one need to farm up the mats on each character so that multiple staves can be created, or is one staff (per difficulty, etc.) enough, since it can be used by any character you have?
Yes, the Staff of Herding is tied to your Account, not your Characters.
Which means you can pretty much use it for any character you have in your Account as long as you have it in your inventory when you visit the Spirit of the Cow King.
Also, i'm pretty sure one Staff can open all the Whimsyshires below the current Mode Level, which is to say a Nightmarish Staff of Herding can also open a Whimsyshire on Normal. Like backwards compatibility.