Assuming Monet could have pressed the button, she would have died right? I genuinely think that Money would have died from the stab wound, but even if for some reason she did not, if she awoke she would have pressed the button to destroy the island. She respected Doflamingo a ridiculous amount. If she was knocked out for an amount of time, after she came to her senses why would she not press the button?
If she came back to Dressrosa after failing to follow through on an order, it would not make sense. We know that the island was not completely destroyed, as the marines went back to the island to save all of the people who were affected by the poison. Therefore Monet is not alive.
As for the rabbits, maybe it is about her fruit being reborn. As for the genuine emotion, she was an interesting character who most likely genuinely felt bad for the children and did not want to hurt people unnecessarily, but she had a stronger desire to follow through on her captain's (doflamingo's) orders.
Also Law had planned out the whole idea with the handcuffs for months, he would have definitely been smart enough to write a note. The details about her are definitely ambiguous, but she is not alive.
It is possible that the G-5 Marines captured her or that Kuzan is with her at the moment inquiring about Doflamingo and the man he is from her. Or Monet could still be unconscious - it has only been a day or so since the events at Punk Hazard ended right? Monet may very well appear at some point later in the Dressrosa Arc if she intends to make up for her failure by helping save Doflamingo.
Anyway I am still of the opinion that the shallow stab by Caesar only managed to penetrate the flesh surrounding Monet's heart and not go deep enough to actually hit her heart (there is a block of flesh missing in her chest that surrounds her heart and as we know flesh does bleed).
And as we can see from this panel, the 'block of flesh' doesn't seem to be releasing blood anymore and from the lack of blood coating the weapon used by Caesar shown, the stab is implied not to be very deep.
The resulting force from Caesar forcing his hand down onto the 'block of flesh' likely caused an impact on the heart of Monet’s which resulted in knocking her out and Monet's weakened state could have attributed to her coughing up blood. Law was knocked out (and coughed up blood) when Caesar applied force to his heart when he was "caught" so Monet getting knocked unconscious because of the force applied as opposed to her heart being penetrated doesn't seem that unlikely.
And regarding the note Chopper received, if Law had been behind it, why did Oda make that scene so ambiguous as to include the possibility that it could have been Monet? The fact that Oda intentionally had Monet exit the focus of the panels seemingly indicates something.
However, my point is she could not come back in the dressrosa arc because if by some miracle she woke up, she would not hesitate to press the button immediately right? Even if she woke up three days after the incident, her orders were to press the button and she would do so as soon as she could killing her instantly.
You are very right to say that there is the very very minute possibility that she was captured by the marines. I do not think that is very likely as they were more focused in rescuing their colleagues.
Your argument does leave a fraction of doubt in my head. I guess there is a very small possibility she could be alive.
Activating the poison gas weapon with the Strawhats and Law off the Island would be quite pointless and senseless to Doflamingo now, especially with Smoker already having made his report to the Marines. The cat is out of the bag, Monet activating the poison gas weapon on Punk Hazard now wouldn't amount to anything for Doflamingo, knowing that she wouldn't activate the poison weapon, unless she wants to die.
1. She would have no way of knowing that they were off the island as she would have no way of knowing the length of time she had been unconscious. Furthermore, there was no view of the outside world from the room she was in.
2. She wouldn't go back to doflamingo a person she respected so much. He even said the words die for me. She was a character who loved DD too much to not follow through.
Couldn't she just use her Den Den Mushi to contact Doflamingo or anyone of the Donquixote Pirate to find out the current situation? Just like how Doflamingo forgave Baby 5 and Buffalo for failing to retrieve Caesar, I'm sure he would forgive Monet once he learns of the exact details that happened.
Anyway if she knew she was unconscious, which would be quite obvious when she wakes up (if she is alive), she would want to gather information concerning the events that had transpired, not doing so would be so unlike the shrewd character she had been fleshed out to be (thinking before acting).
As for going back to Doflamingo, I can see her going back to him if it means being of use to him, senselessly killing herself on Punk Hazard would go against the "die for me" statement Doflamingo made to her because she would die but not for him, she would just die...
I think this is a fundamental disagreement on the tone of Doflamingo's orders.. I think it was mostly clear out the island I have no use for any of you any more.
Also, Monet is sensible and wary. If she woke up, she wouldn't just blindly press the button. We saw when she collapsed that she knew her heart had been stabbed, so she would be conscious of the possibility that she was out for quite a long time. She wouldn't want to waste her life if she knew there was still a chance that she could be of more use to Doflamingo, so she would check to make sure pressing the button was still the best way to do that and she'd find that it wasn't because everyone had gone.