1) For your third interpretation I have at least one relevant experience of my own - "factor systems" (some) number theorists deal with when talking about central simple algebras over number fields did contribute to the development of homological algebra in general, and learning that bit of number theory somehow expanded my homological algebra horizons a bit.
2) Sometimes, it's hard to say where the border between number theory, commutative algebra, and algebraic geometry lies. Around that "triple point" there should be many potential answers.
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