There are several phenomenon that require the vacuum fluctuations to exist, the Lamb shift is one example. However, it seems that this is a bit independent of the question of the vacuum energy. As stated previously, the vacuum energy can be renormalized to be zero, but this does not remove the vacuum fluctuations. Rereading Jaffe's abstract it seems I should have mentioned that he is more specifically focusing on the proof of the vacuum energy, not the existance of both fluctuations and energy. I do not know if there is anything that conclusively supports the vacuum energy.
What do you mean by space though? The vacuum fields require four-space. A common method of describing the vacuum is to assume that we have a cavity with periodic boundary conditions. The true vacuum state is assumed to be the modes of this cavity when we take the limit of the cavity's size to infininty. In this way, we have assumed an infinite Minkowski space. Well, Minkowski space may be a bit of a stretch as I believe that most derivations do not strictly assume Minkowski space but obviously the vacuum can be expressed relativistically.
I am not sure what you mean by energy of spacetime or energy being spacetime. These are concepts I have never heard of.