What do you get when extracting Flames of Rebirth?

You say rebirth flame, that's not the one for meister flames at all is it?

They also require an alchemist with a cube extractor to extract right?

The Rebirth Fragments are required to craft Rebirth Flames (5 for a Powerful, 10 for an Eternal). They're basically the flame equivalent of Cube Fragments in this regard.
For extracting, an alchemist can extract it either by themself (must be Master Craftsman or Meister to do so) or share the ability with a Scroll Extractor. But with this new Profession update, I'm not sure if Blacksmiths/Jewelists can do so as well.