If a combat showdown begins with me having an assault unit at the battlefield and my opponent having 1 other unit at the same battlefield and their unit is removed during the showdown, does my unit retain assault even though it is alone at the battlefield? Does the showdown stay a combat showdown?

Yes to both — Assault is tied to the Attacker designation, not to whether an opposing unit is still sitting across from it.

Assault is "While I am an attacker, I have +X [M]," and it stays in effect as long as the unit keeps the Attacker designation (Rule 807.1.d.1). Attacker/Defender designations are only removed at the end of Combat (Rule 466.7.a) or from a unit that itself leaves the battlefield — an enemy unit dying or being bounced doesn't strip your unit's designation, so your Assault bonus persists until Combat ends. FAQ #1195 is directly on point: Assault applies in combat showdowns until the end of combat, even if the defender moves its units out.

Likewise the showdown remains a Combat Showdown once it has opened as one. Removing units mid-showdown doesn't end it or revert it to non-combat — it still closes only when both players pass consecutively, and it runs through the Resolution Step (FAQ #757, #776). Practically, since only your unit remains at the battlefield, the Combat Damage Step is skipped (Rule 465.1 requires both Attacking and Defending units present) and you win the Combat in the Determine Combat Result task. Note the flip side: if your Assault unit is the one that leaves the battlefield, it loses the Attacker designation and Assault falls off immediately.