Can Kharox be empowered mid-showdown and play the stolen unit during showdown?

Partly — Kharox can't empower itself mid-showdown, but if something else empowers it there, the trigger and the stolen-unit play do resolve during the showdown.

Empower is an activated ability with no [Action]/[Reaction] tag, so it's locked to your Main Phase in a Neutral Open state — never during a showdown (Rules 827.1, 145.2). If Kharox gets empowered by an outside effect while a showdown is ongoing (a card that empowers it, a triggered ability, etc.), the "When I become [Empowered]" ability is triggered, not activated, so it goes on the chain mid-showdown fine (Rule 383.3.c), and the reflexive "choose a unit in their trash and play it" is a Limited Play done as the ability resolves — it doesn't need Action/Reaction timing (Rule 420.3).

One placement limit: the unit enters at your Base or a battlefield you control, so it can't be dropped into a contested battlefield to join the fight already in progress (Rule 355.2).