If I use Heedles Resurrection to kill Brynhir during an enemy's turn (my reaction), can I play him right away from my trash? Since I paid using Brynhir, technically I can use it from my trash?

No, you cannot use Heedless Resurrection to kill Brynhir Thundersong and then immediately play that same Brynhir from your trash.

According to the rules of game timing and targeting, this interaction fails for the following reasons:

The Sequence

1. Play the Spell: You announce you are playing Heedless Resurrection.
2. Make Choices: You must choose a target unit from your trash to be resurrected. At this moment, your Brynhir Thundersong is still on the board and is not in your trash, so it is not a valid target.
3. Lock Targets: Targets are locked in during Step 2 (Rule 355.15). You cannot change this selection once the spell has begun the process of being played.
4. Pay Costs: You then pay the additional cost of the spell by killing your friendly Brynhir Thundersong, moving it to the trash.
5. Resolution: Because your target was already locked in during Step 2, you cannot retroactively choose the Brynhir that just arrived in the trash.

This is a specific limitation of how Heedless Resurrection (and similar spells with targeting requirements) functions; the game engine requires you to declare your target before you pay the costs that would put a new card into the trash.

Sources: RiftJudge FAQ #9523, FAQ #9891, and FAQ #10143.