Is it part of special relativity that mass possessing energy is more dense? I was reading http://www.edge.org/3rd_culture/hillis/hillis_p2.html and it says that a charged battery weighs more than a dead one or a rotating object weighs more than a stationary one (i.e. mass containing energy weighs more than mass that doesn't).
Is this true in special relativity?
Because the energy would be confined to the area of the original mass, wouldn't it become more dense if it is true?
Also, if the mass we are speaking of does become more gravity-sensitive on energy gain, does it actually gain physical mass (protons, electrons and neutrons)??
 A: The reason a rotating object weighs more than a non-rotating one is that weight is a quantity that indicates how strongly gravity is coupling to (interacting with) the object in question.  When the object rotates, its energy increases, and since what determines the strength of the gravitational interaction is energy and momentum of the object, not just rest energy (mass), the weight of the object increases.
This might be confusing at first because Newton's law of gravitation indicates that the strength of the gravitational interaction between two objects goes like the product of their masses and says nothing about energy, but this law is only approximately true, and general relativity tells us otherwise.
The "mass we are speaking of" as you say, does not change.  The term "mass" in modern physics is used to describe what people often call "rest mass."  The number of atoms in a spinning object does not change.  This is true both in special, and in general relativity.
A: 
Is this true in special relativity?

Well, it is a consequence of special relativity. So yes, I guess.

Because the energy would be confined to the area of the original mass, wouldn't it become more dense if it is true?

It is not true in the general that the volume of the mass remains the same. There are shortening effects as well. For example in the linear case: if you just consider a linear mass with rest mass $m_0$ and rest length $l_0$, then its linear mass density is $\mu_0=m_0/l_0$. And if the mass is now moving along its length at velocity $v$, then its new mass is, $m = \gamma m_0$ and new length is $l = l_0/\gamma$, where $\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ is the Lorentz factor. Which makes its new mass density: $\mu = \gamma^2 m_0/l_0$. So the mass density has increased by a quadratic factor. Since energy is simply $mc^2$, the energy density has also increased quadratically. 
But for a linear mass but with some other shape (say in the shape of a U), it gets more complicated. Say its moving along the $x$-directions, then its length along $x$ will decrease but its length along $y$ and $z$ will stay the same. So the total length might change in some complicated way. Other issues in higher dimensions is that the Lorentz factor for the mass will use the total velocity while the Lorentz factor for length along each direction will use the velocity along that direction. I have no idea how to do the calculation for a rotating body. A rotating sphere would be a really interesting calculation to do. 

Also, if the mass we are speaking of does become more gravity-sensitive on energy gain, does it actually gain physical mass (protons, electrons and neutrons)?

It doesn't actually gain more protons, electrons, neutrons. What happens is that the fundamental particles themselves get 'heavier'. Are you wondering whether energy gains might contribute to gravitational mass but not to inertial mass (I guess that's what you are referring to as physical mass)? As far as we know, those two things are the same, but we don't know why yet. 
