It is possible to patent something, and then make the patent publicly available for use free of charge. Once you have made the information describing the idea public, then you have exactly one year to file in the US, but cannot file anywhere else. No one else should be able to file a patent on your idea from the day that you have made it public (unless they had the same idea earlier).
However, filing patents is expensive. I do not know if I would bother if you do not want to charge anyone to use the idea.
Also, I am not a patent lawyer (nor a lawyer at all).
Regarding your actual university applications: I think that affordability might be as much of an issue as acceptance. Have you run the NPCs? Do you have $60k per year, or $60k in total (I was thinking the latter, but am not sure)? You might want to think about applying to universities that are very selective but offer full need based aid, assuming that the NPC agrees with your family’s sense of what you can afford.
You have done very well up to now. Congratulations!