Now, as for the throw-weight of our nukes, I know very little about yields, and I suspect that real detail here is classified. But, at least according to an article I read in Scientific American in the early 1980's, there appears to be such a thing as a "shaped nuke", just like there are shaped charges of conventional explosives. Assuming such IS actually possible, I suspect we'd use that sort of nuke.
As for the energy transfer, the energetic particles of the bomb itself would transfer their energy to the matter of the asteroid, and since it's vacuum on one side, and rock on the other, the explosive vaporization would be on the side of the bomb blast, producing a massive short-term thrust along the rough line of the original blast. It's all Newtonian physics from there (g)
Actually, much of the energy transfer would be in the vaporization of the surface of the asteroid that is exposed to the blast (via radiation). This vaporization would lead to a "jet" of sorts, nudging the asteroid. 'least dats what I reads...