[verb] one's way {{in | into | to } [target] | through [hindrance/obstacle]}
is a general pattern where you can substitute different verbs, like "force", "claw", "push", "work", "beg", or really anything that is clearly understandable in the context.
It means, roughly, "Attaining [target] by [verb]-ing, thereby creating a path or entry into [target] (or through [obstacle]) for oneself, where a path or entry wasn't available."
The worm cheerfully munched its way right to the center of the apple. [There was no path to the center of the apple; the worm created one for itself by eating.]
She slept her way into the movie industry. [She had sex with directors or producers in exchange for getting roles in films, thereby creating a way into the industry that was not otherwise available.]
The usage "don't you have to push your way in" is clear: "don't you have to get into the game by means of creating an entry for yourself via pushing?"
"in" is correct, because that's the preposition that is used for indicating participation in a game: being "in the game".
"Push" is also appropriate, because it has figurative meanings, one of which is forcing or asserting oneself. (E.g. an overly assertive person is called "pushy" and "pushes around" other people).
Best Answer
Your bold-font phrases are standard informal English. There are many ways to express the idea, e.g., "Here's how it'll work: first A will do X, then C will do Y, then E will do Z, and, finally, G will do the last part."
Explaining the order of things is easier using a chart or just a piece of paper and a pencil. Your English seems good enough that you should be able to handle this without worrying about being understood.