When Orianna transforms a gear into a unit, do abilities printed in the gear's rules-text box trigger and resolve when that gear-unit conquers or holds. For example when she makes Warmog equipment a unit, will its conquer effect trigger?

No — as written, the conquer/hold abilities on Equipment sit in the effect text box (below the Equip line), and effect text is Inactive unless the card is Attached (Rule 724). Orianna's Lady of Clockwork makes the gear a unit gear, which grants it unit properties but does not Attach it, so "When I conquer, buff me" on Warmog's Armor would not trigger or resolve.

The distinction is where the text is printed. The Equip line is Rules Text and stays active (Rules Text is never inactive by default, Rule 723), so the thing can still be equipped later — but the triggered ability in the lower box does not fire while the card is unattached, and Inactive abilities "do not trigger, do not apply" (Rule 721.2). Being a unit doesn't change that; it just means it won't be recalled off the battlefield (Rule 178.1.a.1.a).

Note the flip side: if the equipment is attached when it matters, its effect text becomes active and is appended to the unit it's attached to (Rule 136.2.c) — in that case the host unit is the one that has the conquer/hold trigger, not the gear itself. No FAQ covers Lady of Clockwork specifically; this reads off the Inactive text rules.