If the potential is 0, the gravitational force should also be 0
The gravitational potential is defined to be a thing that, when differentiated, gives you the gravitational force. The relationship is just like velocity and acceleration. The velocity of a ball thrown upwards will stop momentarily, be zero, while the acceleration due to gravity is a constant that is never zero. You should thus see that your argument here is absolutely unphysical.
How does this change the gravitational force equation?
Adding a constant to the gravitational potential changes the gravitational force by nothing.
Can we, and if so how do we define the gravitational potential of a mass (say the sun) to be zero at a finite distance (say 1 light year)?
Take whatever gravitational potential that is standard, and find its value at one lightyear (some negative number), then subtract this value (thus add a positive number) to the gravitational potential everywhere.
I would assume the graph would start low (highly negative), become 0 at 1 ly, and then grow indefinitely large.
It will grow to the same constant positive number that is added, and no more.
There is no physical sense in which we would want to do such a thing. It is extremely natural for the potential energy of things to be zero when infinitely separated; any other value is physically stupid.
This is because the specific value of energy counts in relativity. That is, if you do not respect this natural definition of the potential energy, then you will have to tell everybody how to change the definition of how rest mass is calculated when you separate a proton and an electron far enough at infinity, how their individual rest masses are defined, separately, and what happens when you combine them to make Hydrogen atom. It is way too much stupid work when you can just define the potential energy at infinity to be zero, and never have to think about this ever again. Everything will turn out correct when you use natural definitions when they appear.
