Does Last Rites target when you put it on the chain or on resolution?

You choose the target when you put the ability on the chain (during finalization), not on resolution.

For the [Equip] ability — "Attach this to a unit you control" targets. Per Rule 355.14.b, targets are chosen when the spell or ability is finalized on the chain, and per FAQ #4550, equipping specifically counts as targeting. So at the moment you activate Equip, pay the cost, and place it on the chain, you pick which unit it attaches to. That target is locked in — your opponent can react, but you don't get to pick a different unit on resolution.

For the triggered effect ("When I conquer or hold, you may play a unit from your trash") — this doesn't target at all, so the question doesn't apply there. Per FAQ #10819, you decide whether to play a unit when the trigger resolves, and then you play that card through normal procedures (choosing it from your trash as part of playing it).