In combat, I'm attacking with 2 units. One of them has an equipped Last Rites. I win the combat, but my opponent kills the unit equipped with Last Rites. Do I still trigger the conquer effect of the gear?

No, the conquer effect of the gear will not trigger.

According to RiftJudge FAQ #3012, conquer effects require the source of the ability to be present on the battlefield at the moment control is established (which occurs after the showdown or combat ends). If the unit equipped with the gear dies during the combat, it is no longer on the board to "see" the conquer event, and therefore its abilities cannot trigger.

Sequence of Events:

1. Combat/Showdown occurs: Damage is assigned and resolved.
2. Unit Death: The unit equipped with Last Rites is killed during the combat resolution.
3. Cleanup: The unit leaves the board.
4. Control Established: Control of the battlefield is determined after the combat ends.
5. Trigger Check: Because the unit (and the attached gear) is no longer on the board, the "When I conquer..." trigger condition is not met.

Death triggers resolve during the resolution step of combat, before the battlefield is officially conquered. Since the gear is attached to the unit, it is removed from the board along with the unit when it dies, preventing it from triggering when the conquer event finally occurs.

Note: I could not fully verify the reference to FAQ #7489 against the retrieved rules.