There's a board of Hwei on player 1 side and an Ashe on player 2 side. Player 2 activates Cull the Weak. Player 1 responds with Flash. Does player 1 get Hwei movement trigger?

Yes — Flash counts as a move, so Hwei's "When I move" trigger fires and resolves before Cull the Weak.

Chain order matters here: Flash is on top of Cull the Weak, so Flash resolves first and moves Hwei from his battlefield to base — a Move per Rule 441. That move generates Hwei's trigger as a new Pending item on top of the chain, so he draws 1, discards 1, and gets his type-based bonus before Cull the Weak ever resolves. Then Cull the Weak has each player kill one of their units — Player 1 can choose Hwei or a different unit, and Player 2 must kill one of theirs (Ashe doesn't interact with this at all). The move trigger doesn't require Hwei to survive; it just needs him to actually move.