They will certainly have an reaction--we'll get a council scene in chapter 1 or 2 of season 2. It will definitely have an effect, and she'll be put on even stricter watch--but as much as they want to attribute it to her, it could so easily be him catching something in the dungeons and croaking.
We know it's not that, and the council knows it's not that, but his death would be attributed to being in the dungeons.
It doesn't change the restrictions though. And it will make Mertha both afraid and even more angry at her, because, as you said, it happened under her watch.
Otto will be annoyed but pragmatic--she is married now and theirs. And I think a part of him will be impressed.
Larys? I have no idea, but I imagine he'd be amused.
Gwayne may be shocked, surprised, even a little off put but at the same time, I think he'd understand why she did it--he has a bit of his father in him, so he can be pragmatic as well, while being sympathetic.
Aegon will make some japes--both at Dae and Aemond, though he's sleeping off his hungover when the council meeting is held.
I think Daenera will talk with Edelin about it, about why she did what she did. Edelin likes Daenera--Dae is kind to her and treats her well, almost an equal and she's been relatively honest with her, so I also think Edelin would outright ask her about it.
As for Finan, I think he knew when he provided the berries that there was a purpose behind it. He will feel a sort of guilt or maybe even shame, but at the same time, he understands because he's seen what happened to the other men, what would undoubtedly have happened to Patrick, and he knew where he was, the boy wouldn't have survived for long in those dungeons.
Patrick's death is a sacrifice and a mercy--but his death would have been inevitable even if Daenera didn't make this choice. He already had the nose around his neck, she just eased him into sleep.
The first chapter will deal with the immediate aftermath and all Daenera's thoughts and emotions about it. She will swing from justifying it to thinking herself selfish, being sad and then callously ruthless. I think its something that will stay with her for a long while.