Rampage and Challenge make Might and damage. when it get healed?

No — damage from Rampage and Challenge does not heal right away. It stays marked on the units until one of the two healing windows: a Combat Cleanup (after a Combat Showdown where opposing units actually fight) or the end of the turn (Rule 143.3.b).

Neither spell triggers healing. Rampage and Challenge are spells that make units deal damage equal to their Might — that's spell damage, not combat damage, so no combat cleanup happens when they resolve (FAQ 4257). If you use them outside of combat, the damage persists on the board until the next combat showdown ends or the turn ends, whichever comes first.

One note: healing is global — when any combat showdown ends, all units on the board heal, not just the ones that fought. But conquering an empty battlefield is not a Combat Showdown, so that won't heal anything (Rule 466.1).