If my phoenix is targeted by an enemy's Falling Star and I Deathgrip it and play it again since I've killed it with a spell, does Falling Star fizzle and target becomes invalid or does my phoenix die anyways?

To answer your question: If you use Deathgrip on your Immortal Phoenix in reaction to an enemy's Falling Star, your Immortal Phoenix is removed from the battlefield (killed by your Deathgrip) before Falling Star resolves. Consequently, Falling Star will "fizzle" (or more accurately, fail to resolve its instructions) regarding your Immortal Phoenix because it is no longer a legal target.

The Sequence

1. Falling Star played: Your opponent plays Falling Star targeting your Immortal Phoenix (either once or twice). Falling Star is now on the chain.
2. Deathgrip played: In response, you play Deathgrip (which has [Reaction] timing), targeting your Immortal Phoenix to kill it.
3. Deathgrip resolves:
   • Deathgrip kills your Immortal Phoenix.
   • Immortal Phoenix is moved from the battlefield to your trash.
   • Immortal Phoenix's triggered ability ("When you kill a unit with a spell...") is added to the chain because you were responsible for killing a unit with a spell.
4. Immortal Phoenix ability resolves: You choose to pay the cost to play Immortal Phoenix from your trash back to your base (entering exhausted).
5. Falling Star resolves:
   • Falling Star checks its targets. Since your Immortal Phoenix is no longer on the battlefield, the target is now illegal/invalid (Rule 359.3.e.2).
   • According to Rule 359.3.e.5: "If any of the spell's targets are no longer legal, those targets are unaffected by the spell as it resolves."
   • Therefore, Falling Star cannot deal damage to the Immortal Phoenix that was initially targeted.

Important Nuances

• Targeting: Because Falling Star requires you to choose targets when it is placed on the chain, those targets are locked. If that specific unit is no longer in the zone or state required by the spell when it resolves, it is an invalid target.
• "Fizzling" vs. Invalid Targets: The entire spell does not necessarily "fizzle" if it has other legal targets. If the opponent chose to split Falling Star's damage between your Immortal Phoenix and another unit, the damage instance targeting your Phoenix will be ignored, but the instance targeting the other unit will still resolve normally.
• Immortal Phoenix Trigger: You are able to return your Immortal Phoenix because Deathgrip is a spell, and you controlled the spell that was responsible for killing the unit (as supported by FAQ #9374 and FAQ #4668).
• Not the same object: Even though you played Immortal Phoenix again, the rules treat the unit currently on the board as a new instance/game object compared to the one targeted by Falling Star (Rule 359.3.e.4).

This is my interpretation based on the provided rules for targeting and chain resolution. No official FAQ exists for this specific interaction between Deathgrip and Falling Star.