So does winning competitions like Mathcounts, Science Bowl, or Science Olympiad or Dupont Essay Contest help your college application?
Yes. No guarantees, of course, but those awards would certainly not be considered a negative.
Yes, very much so. Of course, on their own, they aren’t all that helpful, but a national level or a state level (or a regional level - a Chicago, LA, or NYC level award, for example) award can make an already competitive application stand out, and can likely make a borderline application competitive.