Does the 4 damage from Ruined Rex get added onto the Showdown Might calculation? (i.e. does my unit fail to conquer if the damage from Ruined Rex + the Might total of the defense is higher than my Might?)

No — Ruined Rex's Deathknell damage is never added to the Showdown Might calculation, and it cannot cause your unit to fail to Conquer in that way.

Damage and Might are separate values (FAQ #2478, #3817). The Deathknell deals 4 damage to an enemy unit — it does not increase the enemy's Might. The Might totals that determine who wins the Showdown are summed during the Combat Damage Step (Rule 460.2.a–b), which happens well before the Deathknell trigger even resolves.

The full sequence:

1. Combat Damage Step — Might totals are summed, damage is assigned and dealt. Ruined Rex dies here.
2. Combat Cleanup — all surviving units are healed (any combat damage they took is removed).
3. Deathknell resolves — dealing 4 damage.
4. Combat result is determined.

So the Deathknell's 4 damage lands after the Might comparison is already done and after the healing step. It can kill a weakened unit and potentially affect which units remain at the battlefield for the result check — but it never factors into the Might totals that determine who "wins" the Showdown.