Faye's very unlikely to be dead. Alcohol poisoning kills you by knocking out the brain stem, which stops the breathing reflex. That's why you're advised to check the breathing of anyone drunk and unconscious: if they breathe normally, there's a fair chance they're just sleeping; if they're not, dial your local emergency number NOW. (But don't take their breathing as gospel: try to wake them up, which is what Marten does in panel 1. If the say "Get lost", or anything intelligible and relevant at all, they're fine. If you can't make contact, again, call the local emergency number, which is what Marten did. Well done. By the book.) Faye stopped drinking some time before the last panel, probably a couple of hours or so, meaning her blood alcohol isn't going up any more, so she should be fine. And if she isn't, they'll put her on a ventilator and she will be fine. Except that she'll have a mother of a hangover when she wakes up.
All of this assumes that Jeph isn't messing things up further by Faye having a pre-existing medical condition like an aneurysm or whatever, but that's a bit like "and then the ambulance careened off the edge of Grand Canyon on the way to the hospital", which, er, No.
And she punched Marten simply because the weirdest things can look like a good idea when you're drunk. Trust me on that last one, said the bloke who once got himself locked up under a particle accelerator.