Wow, there has been a lot there.
Junipurr Determining a gravity assist trajectory is known as a NP problem, which means there's no easy way to solve it. It's generally done with a computer. However it's possible to calculate relatively easily (though it's approximative) the deviation obtained from a gravity assist. The trajectory around the body is an hyperbola, which equation is:
r = p/(1+e×cos(θ-φ))
where p is the orbital parameter, e the eccentricity, and φ the argument of periapsis.
e<1 means the trajectory is elliptic
e=1 means it's parabolic
e>1 means it's hyperbolic
For an hyperbola, the deviation angle is equal to:
2×acos(-1/e) - π
To use those formulas you have to calculate p and e from ingame data. I have some formulas for that, but I'm on holidays right now, I don't have my notebook with me. But you can probably find them easily with the following formulas:
periapsis = p/(1+e)
1/2 × v^2 - μ/r = -p/(2×(1-e^2))
You have to know the periapsis, and the speed at a given radius. I usually use the speed at the SOI level for this, but it works with any point. Then you have 2 equations, and 2 values to determine, p and e. The second one is very general while being quite simple by the way. It can be applied in a lot of situations, that's often a life saver. If you need φ you'll need a protractor. There are some protractor apps that can help you to get an angle from a screenshot.
That's relatively simple and predictable until now, but in practice this is still approximative. Firstly, because the deviation formula is exact when you consider it for an infinite radius. But the approximation is rather good in practice. But there's also another problem, during the gravity assist, the planet still rotates around its parent body, and the planetary configuration changes during the time you perform your swingby. This slightly changes the deviation angle. In practice, you have to evaluate the angle by which the planet has moved, and to substract it if you made a prograde slingshot, to add it if you made a retrograde slingshot. That's the reason why retrograde slingshots are slightly more efficient in practice.
But to evaluate that angle is more difficult, because it depends on the planet speed (which can easily be found because the orbits are circular), but also on the time you spend being in the SOI. And for this one you need the Kepler equation. I can't remember the exact formula, but you can find it on the internet (be careful, there's an equation for elliptical orbits, and another one for hyperbolic orbits).
With all of this you can calculate the result of a gravity assist quite precisely, but it's still an approximation, you would have to check the actual result ingame, and overall that's already a tedious process.
I'll try to help you, but as you see, it's rather a long process.