Where from is the distance calculated in Newton's law of universal gravitation? In the equation for universal gravitation $(1)$ between two objects, where from is $r$ calculated? From the surface, from the center? Also, are the objects assumed to be particles in this equation or could the be multiple particles like molecules?
$$F=G \frac{m_{1} m_{2}}{r^{2}}\tag{1}$$
 A: Strictly speaking, this equation for the force due to gravity only holds between point-like objects. In the case of a point-like object, the notion of "distance" between them is simply the distance between their positions.
For extended objects, things get a little more complicated. Really, if you had an extended object (a sphere, cylinder, amorphous blob) you would need to imagine breaking the object into many many tiny pieces of mass, compute the force due to gravity caused by each small piece, and add them all up (as vectors! not all the forces will point in the same direction, see for example tidal force).
The only exception to this is the gravity due to a sphere...in this case we can use the distance from the center of the sphere to compute the force due to gravity. But this is a very special result that comes about from the "breaking apart" that I described above...it just so happens that things work out the same as if we had just used the center of the sphere. For example, see here.
A: The object in this law assumed to be points (so the distance is just distance between points). Any two object that are very small compare to the distance between them (no matter from which points on the objects you measure it) can be considered as points.
So, for example if you have a two rigid molecule that are small compare to the distance between them, then you can use this equation to calculate the force between them.
A: The distance between two objects is the distance between centre of gravity of this two objects.
