1) With quizzes, their scores could be automatically uploaded to the leaderboard via script (see #3). Perhaps completing a task reveals the answer to an otherwise impossible quiz question?
2) What types of themes will the challenges be? Application should probably depend on that.
3) The players submit their scores to a special server that runs a grading script which automatically updates the leaderboard based on # of resubmissions and score. I know this can be done because it's essentially how my homework is graded, though I don't know how to do this myself. Note that since this game is new, you'll likely run into all sorts of problems and so players will probably want to contact you outside of Friday challenges regardless.