Is there a term for the area of the big bang to distingush it from the bulk universe?
As far as I know no term like that exists, because no physical distinction is possible according to current mainstream beliefs- the bulk universe is simply the big bang "area", stretched out over time.
Within cosmology, we usually use the term "Hubble Volume".
No, the Hubble volume is only equal to the observable Universe for a particular set of cosmological parameters, which are not those we find for our Universe.
To answer the OP, I'm not exactly sure what you are asking, but I think the best answer would be 'the observable Universe' which is what we conventionally call the patch of the Universe we can in principle see given the age of the Universe and the expansion history (which tells you how 'far' photons can travel from the Big Bang until now).
As MikeyW said, we have no reason to beleive the regions beyond this are substantially different in nature, and in terms of the models we have for the Universe we assume that this region has on average the same kind of properites as the part of the Universe we can see, but of course since we can't see it we don't know for sure.
No such area. Big bang was everywhere you look, just as MikeyW said.
Well, not exactly. The Hubble volume is a word which is still bandied about to describe our local universe, though we understand it's not specifically talking about the universe out to the limits of our vision. The original poster didn't specifically say the observable universe.
As far as I know, there actually aren't any cosmological parameters for which the Hubble volume is exactly the size of the observable universe. But in De Sitter space it is the volume inside the future horizon.
Separate names with a comma.