If I use Full the Weak and kill my own Immortal Phoenix, can I use the Immortal Phoenix that got killed to come back?

Yes, you can play that same Immortal Phoenix from your trash.

According to the official FAQ and Rule 376.2.c.1, this interaction works because the Phoenix is moved to the trash during the resolution of the spell, and its trigger condition is evaluated immediately after the unit is killed.

The Sequence:

1. You play Cull the Weak.
2. As the spell resolves, you choose to kill your Immortal Phoenix.
3. The Phoenix is moved from the board to the trash (Rule 415.1).
4. The spell finishes resolving. At this point, the game checks for triggers. Since a unit was killed by a spell and the Phoenix is now in the trash, its ability triggers.
5. You may then pay {1} Energy and {1} Fury Power to play it from the trash.

Relevant Rule:

Rule 376.2.c.1: "If a Game Object with a Triggered Ability that is active in a specific zone, it is evaluated and subsequently triggered if it enters that zone at the same time that its Trigger’s condition is met. Example: Immortal Phoenix... This ability triggers if Immortal Phoenix is in your trash immediately after you kill a unit with a spell, even if the unit you killed with a spell was that Immortal Phoenix."