If I react, killing or taking out the board the unit selected with a "Death from below", my opponent, owner of the spell, can play it again? I think not, because "Death from below" checks the might of the unit killed with it.

You are correct. Your opponent cannot use the recast effect of Death from Below if you remove the targeted unit from the battlefield before the spell resolves.

According to RiftJudge FAQ #9402 and Rule 359.3.e.12, the interaction resolves as follows:

Sequence of Events

1. Play: Your opponent plays Death from Below and targets a unit on the battlefield.
2. Response: You play a reaction that kills or removes that unit from the battlefield.
3. Resolution:
   • When Death from Below attempts to resolve, it checks its target. Because the unit is no longer on the board, it is an illegal target (Rule 359.3.e.2).
   • The primary instruction ("Kill a unit") cannot be executed because the target is illegal (Rule 359.3.e.5).
   • The spell then attempts to check the target's Might to evaluate the condition: "if it had 3 [S] or less." Because the target is no longer on the board, its Might is treated as "null" (Rule 359.3.e.12).
   • Since the condition cannot be evaluated (it returns null), the "you may play this from your trash" effect does not trigger.

The spell resolves, but because the target is illegal, it fails to perform its primary instruction and cannot fulfill the requirement needed to trigger the recast effect.