Hello Tris, you have asked a big question… with no simple answer. If doing as you suggest, digging to remove all roots seen and then composting, for planting next spring, you will need to pull horsetail regularly for the life of the asparagus plants, even though it will be somewhat (how much I don’t know) weaker than at present.
For less horstail, you could cover a whole half of your allotment with balck plastic, total light exclusion, until next winter (2013), after spreading compost/manure and any other organic matter on the ground. Then plant spring 2014 and use the remaining ground for other veg. First taste in 2016! (Covering a larger area is to lessen the problem of horsetail creeping in from the sides)
As to how many plants, a dozen to fifteen, at 2×3 feet, should give plenty of meals after about five years.