My favorite answer to #2 and #3 is Kostant's "Find the highest root game", which is written up in detail in section 5.4 of [Balázs Elek's notes on reflection groups][1]. It is not hard to show that all plays of the game (from all starting positions, i.e., simple roots) take you through all the positive roots.

  [1]: http://pi.math.cornell.edu/~allenk/elek_reflection_groups_2016.pdf