With all the important caveats (about getting too emotionally invested in highly selective schools) out of the way, let me say that I think you have some very competitive aspects to your application - portraying a mature, independent, highly engaged, and personally resilient young woman.
Your US Rep internship, and the specifics of tasks there, your crucially important caregiver tasks, the depth of your engagement in your Africa Service Project (including the funds-raising aspect!), and your various after school/vacation jobs you held, would already be impressive on their own - but even more so when combined with athletic achievements in multiple disciplines and in light of the personal challenges you had to overcome.
I think there is enough substance for an admissions officer to see your excellence, when there is no standardized tests and class rank available.
Of course, no matter how much a good fit you would be for Barnard, the ultimate selection from among the too many top applicants into a class are not a reflection on your own application, but what geographically, socially, economically, and otherwise diverse class a college is wishing to “shape” that year.
Therefore, there is no realistic way to assign an individual probability for schools like Barnard - other than that >90% will not be admitted. But wishing you good luck that all the stars align in your case.
PS - you do realize that you can apply to Columbia College in parallel? If by chance you are only admitted at CC, but not BC, you’d still be able to use the Barnard campus, dining rooms, and facilities (other than the health center), join the same clubs, attend many of your classes at Barnard, and (after the first year) even share housing with BC students. My daughter was a Barnard student but her friends circle was from both colleges.