Dora's pretty much in a no-win situation, both in-universe and with a significant portion of the audience. It makes sense in-universe, because she's had to take harsh action against an employee who was also a close friend. The firing was justified - as she notes, it's what any other employer would have done - but logic and rationale is in conflict with her emotions and compassion. She was right to do what she did, but that doesn't make it any easier, or make her feel much better about it.
On a meta level, it's understandable too. She's a divisive character with traits that are going to rub the audience the wrong way. She's certainly erred before, though that doesn't mean she's automatically wrong about everything else; at the very least, there's the old adage of a stopped clock being right twice a day. Still, not every reader is going to be able to cut her some slack, because her past actions have inspired dislike (or at least annoyance) with a part of the audience. You could argue that some people take it a bit too far with their reactions, but it's still an understandable point of view.
All the same, I think the goalposts are constantly moved when it comes to Dora: "Faye broke the rules, but should've been given a warning and shown more compassion...OK, she was already warned and Dora's policy has been quite clear, so the firing is probably justified, but Dora needs to show compassion...OK, she showed that she understands Faye's going through a rough time, but she said the wrong thing when she mentioned insubordination (dialogue that I think was taken out of context, incidentally)...alright, it seems she really does care about Faye after she was rushed to hospital, and the stuff about the insurance is a nice gesture, but wait, she's really terrible for asking Hanners to pass on the message and stressing her out...yes, the firing is understandable, but how dare she mention it to other CoD employees?"
And on and on it goes. Dora can't win for trying, because no matter what she does, there's something wrong with the way she does it, or she didn't say exactly the right thing.
I don't think it was inappropriate to pass on a message through Hanners. She was very concerned about Faye, but knew that she wasn't exactly welcome at her bedside. I don't think it's inappropriate that she's talking to Penelope about this. It's a pretty informal place of work, with a group of employees who know each other fairly well. She's not divulging anything that isn't true (Faye was drunk at work, and she was fired for it), nor going into detail about Faye's problems. From a legal standpoint, I don't think there's much of a case against her at all.
That's something else I've noticed: the prospect of legal action being taken against Dora keeps coming up. A flimsy case aside, it seems like there are people who really want to see her suffer or be punished for firing Faye. Again, kind of understandable given she's a divisive character, but extreme, and a tad silly. She shouldn't be punished for doing, as she acknowledges, what any other employer would do. Furthermore, she is already suffering, because she feels terrible about having to fire a friend.
It is fair to wonder whether she could've been more supportive of Faye, as one of her closest friends. However, it seems they don't socialise as much as they used to, and with Faye's sarcasm and abrasiveness, as well as her secrecy and unwillingness to face unpleasant realities (she avoided the issue of Angus possibly having to move away as much as possible), it's asking a lot for Dora, or Marten, or anyone else to know the extent of her problems. As TheEvilDog pointed out, no one else is ultimately responsible for Faye, either; she may have "hella issues", but she's an adult, and she has to take responsibility for herself and her actions. Having compassion for Faye and holding her accountable for her actions are not mutually exclusive, and her problems are not a Get Out Of Jail Free Card, absolving her of any responsibility or wrongdoing.