I've already ruled this one way at my table, but I'm curious what others think about this situation:

So evil doll swaps with the character, good times. The stat block is clear what happens if you kill the doll body (both die). What happens if the character's body dies? And more importantly, what happens if the character's body dies and is then targeted with healing or a resurrection spell? Does the carrionette soul go back to the body? Nothing happens? The original soul goes back?

I've ruled the body is repossessed by the carrionette's soul, but I can see it go several ways.

First of all this is what the basic rules say about hit points dropping to 0:

Falling Unconscious

If damage reduces you to 0 hit points and fails to kill you, you fall unconscious. This unconsciousness ends if you regain any hit points.

To anwser your questions:

What happens if the character's body dies?

The Carrionettes ability says that she gains control over the body and the Charachter gains control over the Carrionettes body. So if the Charachters body drops to 0 HP or dies. Then the charachter becomes unconscious, not the Carrionette.  But the Carrionette consciousness is still in the Charachters body. So RAW the Charachters body is still being controlled by the Carrionette and does not suffer the unconscious condition, while the charachter is unconscious in the Carrionettes body. But you could also see it as the body of the charachter being unable to move anymore (legs and arms cut off or something you name it), meaning the Carrionettes conciousness is stuck in the Charachters unconscious or dead body, while the charachters concsiousness is unconscious in the Carrionettes body. 

What happens if the character's body dies and is then targeted with healing or a resurrection spell? 

The consciousness only reverts back by either casting a protection from evil or good spell on the charachters body or if the controlled body takes damage from the carrionette’s Silver Needle. So, As long as one of these two criteria aren't met, there consciousness stay swapped. Meaning that if you kill the Charachters body with the Carrionettes Consciousness still in that body and you then bring it back from the dead, the Carrionette still controllers the charachters body. 

Long story short, just don't get soul swapped by a Carrionette smh. Hope this helped!

Haha thanks! That's kind of the way it played out and I ruled it, but it's such a weird mechanic (but super fun) and kind of a niche case. Watching them try to keep others from killing the doll though is amazing. :)

No problem bro. It's indeed a pretty weird ability and also pretty vaguely written. But it makes up for it with some interesting character dilemma's. :)