Actually there are bushings for bottle stoppers. You have a size for the standard stainless steel bottoms and one for the mini. They slide over a BS mandrel. These just simply tell you the size of the outside diameter of the stainless steel bases. This again depends upon what madrel (if any) that you purchase. It's been so long now I don't recall where I picked up mine.
Length is personal preference. I've seen some really long stoppers (3-4") personally I think that's crazy long...but hey, whatever sells. I normally cut my blanks @ 2"
I've also just started turning stoppers with a Collete chuck using a dowel to hold the blank. My experience so far with this method is be sure to 'finish' the stopper before removing it. I tried removing and putting it back and it never spins true again.