If the targeted creature doesn’t have first strike, granting it first strike after combat damage has been dealt in the first combat damage step won’t prevent it from dealing combat damage. It will still assign and deal its combat damage in the second combat damage step.

You may target any creature, not just one that doesn’t have first strike. If the targeted creature already has first strike as Lightning Blow resolves, Lightning Blow won’t affect it (since multiple instances of first strike are redundant), but you’ll still draw a card at the beginning of the next turn’s upkeep.