If I have Spirit Wheel and then play Cull the Weak and kill one of my units with it, have I chosen a unit and thus be able to draw a card with Spirit Wheel?

No. Cull the Weak does not target or "choose" a unit in the game-mechanics sense — the choice happens during resolution, not when the spell is placed on the chain. Spirit Wheel triggers when you choose (i.e., target) a friendly unit, and Cull the Weak's instruction is a non-targeting resolution choice (Rule 355.10.e). So you don't get the draw off Spirit Wheel.