In situations like this, it is hard to say what the correct course of action is. Faye is clearly in the wrong, but did Dora really handle it the right way? Or, to put it another way: How would I react and respond if I were in Dora's position?
My answer? I don't know. My own life has been a series of slippery slopes with only a couple of reliable tracks found thus far (I haven't hit bottom before, but sometimes one's own psychology can make it feel like you're there). I have never been in Dora's shoes, where I've had to make such a decision.
However, was an on the spot firing really justified? I don't think so. Immediate unpaid leave, with strict conditions that Faye gets help to clean herself up in two weeks or then she's out on her ass? That is a fair call, and is strict without being harsh. You've laid down your terms and conditions, and they have to act, knowing there will be repercussions if they don't.
The current decision, given that it is probably mid-morning, could lead anywhere (and BenRG has laid out a few scenarios that get very dark very fast). Will Faye seek comfort in the bottom of a bottle (which, given she lost her boyfriend, her job and possibly also her best friend in the space of a week, is highly likely)? Will Faye decide that Northampton doesn't have a place for her in it anymore, and follow Angus to NYC (Maybe. Angus didn't really want to split, but Faye made the call and he bitterly accepted that decision)? Or will Faye decide that the world has no place for her in it anymore, and attempt or commit suicide? QC is a pretty good reflection of real life, but it would be absolutely fucking heartbreaking to anyone, who has invested so much of their own life, whether it be the characters in the QCverse, or even us as readers and members of this online community, to accept or even really grasp such an act of desperation. Especially so for the people who unfortunately have been there, and more so for those who have to deal with the repercussions of those decisions.