Physics programming is not related to your goal at all. Game mechanics, such as how high characters jump or how far their weapons reach, are completely unrelated to physics.
These kinds of game mechanics are high-level, often script-based.
What kind of education do you need for this? Virtually none. It is just plugging numbers into scripts. You need only the most basic programming skills to work with scripts, and it sounds as though all you want to do will be done with scripts.
Even for fighting games, it would be set up so that the engine already knows that once you get hit you should go into “Unprotected” mode. Your job would be to tell the engine how long that mode lasts if no hits follow. How long does it take to recover from X attack (how long is your guard down after missing)? How much damage does it do? Which moves can be chained after it to form a combo?
The game designer tells you these numbers and you plug them into a script.
Not very fulfilling, very low pay, not a very bright future there.
Certainly being a physics programmer would be more fulfilling and provide a worthwhile salary, but nothing you have said indicates that that is what you want to do, except the part where you misunderstood its relationship to game mechanics.
The way Mario jumps, for example, is a gameplay mechanic, not physics. How far your character reaches is a mechanic, not physics. Nothing in a fighting game or standard RPG is physics. Any falling or bouncing there is hard-coded gameplay mechanics that can be driven by scripts.
If, after realizing the difference between physics-driven gameplay such as in Half-Life 2 and hard-coded scripted faux physics (or fauxsics as I like to say) found in Mario, fighters, RPG’s, and the types of games you described, you still want to do back-end physics programming, then it should be fairly clear how to proceed.
Get an engine such as Unity 3D, override its physics support, and start writing scripts to make balls bounce off each other with your own scripted physics. Still high-level but directly applicable to the low-level beck-end things you would want to do later.
Just be warned that there is virtually no demand for physics programmers today. I was originally an AI programmer and physics programmer (in fact, L. Spiro Engine started as just a physics engine, only becoming a full engine when I decided I need graphics to draw the objects in my scenes) and another of my coworkers majored in physics at his University.
Neither of us have ever had a chance to do physics programming professionally except for some very basic and short-term (a few days) things. Companies use Bullet or PhysX to meet their physics needs, so getting a job as just a physics programmer is extremely unlikely.