Do units heal after Showdown? Open Showdown and Combat Showdown?

No — it depends on whether the showdown is a combat showdown or not.

• Combat Showdown (Contested battlefield with units from opposing players): yes. After combat damage, Combat Cleanup heals all units in play — every unit everywhere, whether or not it fought (Rule 461.1.a.1).
• Non-Combat Showdown (Contested battlefield where only one player has units): no healing. There's no Combat Cleanup after a non-combat showdown, so units stay damaged until the end of turn cleanup (Rule 344.2).

Separately, all units still heal all damage at the end of each player's turn (Rule 317.2.b), so damaged units won't stay hurt forever.