3) It would be interesting to see examples in the same spirit as how the $n$-factorial conjecture ended up being proved using quite advanced algebraic geometry. Maybe a good candidate along those lines is this result of Kanel-Belov and Kontsevich that uses reduction to characteristic $p$: "The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture", http://arxiv.org/abs/math/0512171