There is a formula, but the math behind it is extremely complicated - far more than my college-level mathematics education is capable of. It needs graduate school level math, and my field is paleontology, not physics, so I never studied it. The "flexibility" that you speak of is the universal gravitational constant,G, which is present even in the Newtonian approximation of gravitational force

(i.e. F =G*m1*m2/r²).

That particular formula is still a very good approximation of gravitational interaction, but there are some complications to the solution in the case of very strong gravitational fields, or over very short or very long distances.

I don't have an easy answer for this one, since I haven't done the mathematics for advanced general relativity.Unfortunately, the equations that are easy to write are difficult to understand, and vice versa. One lesson is clear: if you'd like to understand more Physics, take more Math!

The interaction of classical objects and gravity is the subject ofgeneral relativity and there are several good text books on the subject. (My favorite is Meissner/Thorne/Wheeler -- but it is likely dated now). A suitable introduction is the book Space-Time Physics which needs only a bit of calculus to read. The formula relating the curvature of space time to the mass of an object is the Einstein Field Equation which does exactly that. The problem is that the curvature is in a four-dimensional space -- so the equation can be challenging to apply. However, there are lots of simple solutions for situations like spinning massive bodies (like the earth).