If I use Strike Down to have my friendly Lucian Merciless (with Warmog and a buff from Warmog from conquering) then detach the Warmog, does Lucian keep the buff or not?

Yes, your Lucian, Merciless will keep the buff.

According to RiftJudge FAQ #8468 and #529, when an equipment (like Warmog's Armor) detaches from a unit, the unit retains any buffs it already gained while the equipment was attached.

Here is the breakdown of the interaction:

1. Buff Persistence: Buffs are objects placed on a unit (Rule 702). They remain on the unit until the unit leaves play or the buff is specifically spent or removed. Detaching an equipment does not remove existing buffs.
2. Equipment Status: Once Warmog's Armor is detached by the effect of Strike Down, its text becomes inactive on Lucian. While Lucian will keep the +1 Might buff he already earned, he will not gain new buffs from future conquers unless you re-equip him with Warmog's Armor or another source of that effect.
3. Strike Down Sequence:
   • You play Strike Down, targeting your equipped Lucian.
   • Lucian deals damage equal to his current Might (including the +1 buff from the previous conquer).
   • The equipment (Warmog's Armor) detaches.
   • Lucian remains on the battlefield with his existing +1 Might buff intact.