My opponent controls a Battlefield. I contest it with a Kaisa Survivor. They respond with a Hextech Ray and deal 3 damage to my Kaisa. If I flash out of the Battlefield, will the damage heal? Does combat clean up occur and why?

To answer your questions regarding the showdown and the interaction between these cards:

1. Does the damage heal?

No, the damage does not heal because no damage is dealt to Kai'Sa, Survivor. When you play Flash in response to your opponent's Hextech Ray, Kai'Sa moves from the battlefield to your base. Because Kai'Sa is no longer a legal target when the Hextech Ray resolves, the spell resolves with no effect. Since no damage was dealt, there is no damage to heal.

2. Does combat cleanup occur and why?

Yes, combat cleanup occurs. According to Rule 461.7, combat ends when the designated showdown concludes. Even if no combat damage is dealt or all units are moved away, the showdown was still initiated because you moved Kai'Sa to an occupied battlefield. Per RiftJudge FAQ #8342: "Units heal after a combat as long as a combat was started, even if combat damage is skipped or a unit is moved away before damage is dealt... As long as a combat was started... after the showdown closes and after the combat damage step... all units heal."

Summary of the Sequence:

1. Initiation: You move Kai'Sa, Survivor to your opponent's battlefield. A showdown begins, and your opponent gains Focus.
2. Chain: Your opponent plays Hextech Ray, targeting Kai'Sa.
3. Reaction: You play Flash in response, moving Kai'Sa to your base.
4. Resolution:
   • Flash resolves first (LIFO), moving Kai'Sa to your base.
   • Hextech Ray then attempts to resolve. It sees that its target (Kai'Sa) is no longer a legal target; it resolves with no effect.
5. Combat Resolution:
   • Combat Damage Step: No units remain at the battlefield, so no combat damage is assigned.
   • Combat Cleanup (Rule 461.1.a.1): All units in play (including Kai'Sa at your base) are healed.
   • Result: Since no units remain at the battlefield, the showdown ends in "No Result" (Rule 461.3.d).

Note: I could not fully verify this ruling against the retrieved rules.