As the universe expands, how does it affect the mass of objects and therefore the gravity? I think the mass of object would proportionately decrease as the universe expands, as the amount of matter is constant in accordance to the law of conservation of energy. If I am correct in this, would this entail a decrease in energy, as $E=mc^2$. Also from this, as the mass of objects decrease, would the force of gravity also decrease? And if it does, what does this mean to the outer edges of the universe as it expands, as gravity of the center of galaxies are diluted proportionally and exponentially with the rest of objects in space, what would this mean to the objects within it? Would clusters therefore grow farther apart? Moreover, if they do grow farther apart, does the center of the cluster pull harder, or rather does it accelerate the objects in its orbit as there is less and less gravitational contractions of bordering clusters as they are moving farther apart from one another.
I suppose this is speculation, but I can't find anything online to answer this, or simply I am just not looking in the right places. Or I can just be entirely wrong.
 A: It's actually not true that energy is conserved globally in general relativity (GR). GR doesn't even offer a way of defining how much energy there is in a certain region of space, if that space covers a region of cosmological dimensions. Conservation of energy is a purely local thing in GR.
The mass of objects does not decrease as the universe expands.
It is possible to put together pseudo-derivations of things like the equations that govern cosmological expansion by mocking up things like the kinetic energy of expansion and so on. These are not real derivations, although they may provide some connection to our nonrelativistic intuition.
It is possible to learn a lot about GR without a lot of math, but most books that try to do so are sort of half-baked. A pretty nice book for this purpose is General Relativity from A to B by Geroch.

what does this mean to the outer edges of the universe

There is no edge. This is a FAQ that has surely been covered on this site.
A: This is a comment to :

I think it would proportionately decrease as the universe expands, as the amount of matter is constant in accordance to the law of conservation of energy. If I am correct in this, would this entail a decrease in energy, as E=mc^2. Also from this, as the mass of objects decrease, would the force of gravity also decrease?

Mass is not a conserved quantity for large energies ,  momenta and masses.
The newtonian definition of mass $F=ma$ is conserved, so in our everyday world this is used from defining weights to defining money and control commerce. BUT when one goes to cosmological dimensions, the simple newtonian mathematics of three dimensional vector space to describe observations, no longer holds. Special relativity was invented and managed to to fit  high energy observations.
The m in $E=mc^2$ is a variable that does not obey Lorentz covariance so it does not characterize uniquely matter. It is the four vector length that gives a mass that does not change, called invariant mass . The mass of elementary particles is fixed, and all other masses are a vector result of the constituents, and the mass of the resultant vector is fixed for all frames. It is called the invariant mass for this reason.
As you continue your studies you will learn the mathematics necessary to really understand all this. Here is a link for four vectors.
