1/ Triggering battle ability - Planet Carnath says "trigger the battle ability of any planet in play" ... which brought us to question where this ability is trigered from? From the planet I choose to "copy" or from Carnath itself(especially important when triggering Atrox Prime)?

2/ Initiative - I have an initiative, my warlord gets bloodied, combat round finishes and enemy warlord stays on the planet, oponent gets the initiative, right?

3/ Combat - Firedrake Terminators - does their ability (deal 1 dmg when assigned as defenders) kill an attacker before his dmg is dealt (ie any army with 1 HP)?

1-The ability is being triggered from whichever planet you choose to trigger. Carnath only gives you the choice to do so. In your example, the enemy unit would have to be adjacent to Atrox Prime or at an HQ.

2-No. Once initiative has been determined at the beginning of the combat phase, it does not change in subsequent rounds. This is regardless if the conditions that gave you initiative are no longer present. RRG P. 25 3.2.1 Determine Initiative.

3-Yes, it would if the attacking unit has only one HP overall or one HP left. The Declare Defender step happens before the Resolve Attack step, so the Firedrakes would assign their ability damage before the attacker deals attack damage. RRG P.25, 26 3.2.3 and 3.2.6.

On the answer to #2 - Note that initiative is determined at the start of each battle for that battle, not at the beginning of the combat phase. If a warlord moves between the start of the combat phase and the start of the battle at a particular planet, that can affect who has initiative for that battle.

Other than that, yes. Once initiative for a battle does not change once determined.