How do I put an existing model into a game scene(a level)?
You don’t do anything, just press p to play a game. Unless I’m not understanding what you’re getting to…
Do you mean that you want to use a model from another .blend? If so, use the “File >> Append or Link” option in the menu.
Do you mean that you want to use a model from “Scene 1” in “Scene 2”? If so, select the model and press “Ctrl+L” and select “To Scene…” from the “Make Link” menu that pops up.