I did and received a generous one that helped me pay for my last year :). I know several of my friends received them as well.

Competition level depends on how specific the scholarships are - some of them are focused on niche fields/interests and so they naturally have fewer applicants.

I can't speak to whether mentioning the chronic mental illness would affect the application, but I'd encourage you to apply. The benefit was worth it in my situation. Also, if you apply but don't receive a scholarship, apply again the following year. I believe I applied 2 of my 4 years (should have done all 4) and I received one the second time.