The shape characteristics of gravitational wells given different masses and spread of objects I am

The question is a bit open-ended, so this is more of an introduction to the topic than a total answer - since a full answer involves both potential theory and general relativity.
If we start with gravitational wells in classical mechanics, they have a simple form for point masses:
$U(r)=-<!--\; -\; -->\frac{GM}{r}$
is the distance from the point. They are defined everywhere but the point at $r=0$, and stretch to infinity (where the potential is zero).
If you have several point masses their potentials just add up:
$U(x)=-<!--\; -\; -->G\underset{i}{\sum <!--\; \sum \; -->}\frac{M_i}{d_i}$
where $M_i$ is the mass of the $i$th particle and $d_i$ the distance to the point $x$ where the potential is measured.
One can analyse continuous mass distributions in the same way, by integrating contributions from infinitesimally small particles:
$U(x)=-<!--\; -\; -->G{\int <!--\; \int \; -->}_{t\ne <!--\; \ne \; -->x}\frac{\mathrm{\rho <!--\; \rho \; -->}(t)}{|x-<!--\; -\; -->t|}dt$
$U(x)=-<!--\; -\; -->G{\int <!--\; \int \; -->}_{t\ne <!--\; \ne \; -->x}\frac{\mathrm{\rho <!--\; \rho \; -->}(t)}{|x-<!--\; -\; -->t|}dt$
where $\mathrm{\rho <!--\; \rho \; -->}(t)$ is the mass density at the point $t$. The integral goes over all of space except $x=t$. This way you can (with some effort) calculate the gravitational field around spherical bodies, rings, planes etc.
One of the key results is the shell theorem of Newton: if you have a spherically symmetric mass distribution, the potential only depends on the distance to the center (it is also spherically symmetric). Also, there is no gravitational force from the mass shells outside the point you are measuring from: they still contribute potential, though -- at the center of Earth you would be weightless yet have a fairly low potetial. It does not matter how dense the body is, if you are outside it you will only feel a potential dependent on the total mass.
This also explains your secondary question about the Sun as a red giant. The reason Earth is expected to spiral out a bit is that the Sun will start losing mass though strong solar wind, not that the density will go down.
Further, one can find other ways of calculating potentials. Gauss law is very useful. It states that the gravitational force, when integrated around an arbitrary closed surface, is proportional to the mass inside: $\int <!--\; \int \; -->{\int <!--\; \int \; -->}_{S}g\cdot <!--\; \cdot \; -->dA=-<!--\; -\; -->4\mathrm{\pi <!--\; \pi \; -->}GM$. Note how it implies the shell theorem. One can differentiate this, and then arrive at Poisson's equation
${\mathrm{\nabla <!--\; \nabla \; -->}}^{2}U=4\mathrm{\pi <!--\; \pi \; -->}G\mathrm{\rho <!--\; \rho \; -->}.$
Potentials in empty space where $\mathrm{\rho <!--\; \rho \; -->}=0$ follows Laplace's equation
${\mathrm{\nabla <!--\; \nabla \; -->}}^{2}U=0$
-- they are the subject of potential theory and are sometimes called harmonic functions.
To describe their shapes one can make use of the fact that if $U_1$ and $U_2$ are two potentials that solve Laplace's equation, then $U_1+U_2$ also solves it. So one can add together simple potential solutions (just as in the point particle case) to build more complex ones. There is a vast mathematical theory on how to do this. For gravity usualy the important thing is expansions in multipole moments expressed in spherical harmonics. Since most mass distributions are roughly spherical blobs, these combinations usually produce accurate results with few terms. In particular they can easily handle the distortion from an oblate body (the potential is a bit squashed near the body), or more complex things such as the lumpiness of Earth. Once you have the potential you can for example calculate how it affects satelite orbits.
One of the important results is that higher-order harmonics decay faster with $r$ than $1/r$. They have less effect on the distant parts of the potential. A cubical planet will have a somewhat cube-like field near the surface, but further away it will be nearly like it was a sphere (although they don't use the harmonic method).
Beyond this is general relativity. There is not exactly a gravitational potential in GR (the metric tensor is doing some of the job) although for weak fields one can approximate the classical one and calculate gravitational time dilation. There is also a counterpart to the shell theorem.