I don't think there's any one answer to these, but since it's a FicPsych question, here's my take:

FicPsych would certainly try to rehabilitate an agent who's that far gone, and they do have a very secret, very removed from the beaten path inpatient facility. They would certainly keep a close watch on anyone likely to be dangerous. However, given limited personnel, that number would have to be very small, restricted to only those most desperately in need of their particular brand of care. They are very good at what they do, though, and since they have loads of experience with agents who've lost it, I'd say their chances of rehabilitating most people are pretty good. And if you think it would make an entertaining story, I'd love to see you write it. {= )

As for the second question, again, whatever seems like it would be the most entertaining is probably fine. I don't think there's any precedent for that sort of scenario, and even if there were, it would depend on the circumstances in the fic. That said, simply banishing anything that was summoned ought to do the trick.

~Neshomeh

1. Yes, they'd stay in Medical or FicPsych, depending on their particular needs. Most of the time, agents who go nuts tend to be psychotic rather than just garden-variety traumatized or depressed, probably because the badfic has been re-writing their brains. While it has some things in common with schizophrenia or some kinds of bipolar mania, this condition is not the same thing as either one of those real-world diagnoses. World One doesn't have the facilities to treat badfic-related disorders; the only such facilities are located at HQ (and possibly some other metafictional continua). To have any chance of recovering, insane agents would have to stay at HQ.
   
   2. Depends. The first possibility is that Bloody Mary or the Slender Man would simply be an OC with that name, and dealt with like any other OC. The second possibility is that they would be dealing with the real Bloody Mary. There seem to be characters who have an existence beyond a single continuum of the multiverse. Only a very few are known. Among them are Lady Contrivance and Lord Implausibility, Sekhmet and Imhotep from Egyptian mythology (encountered by Suicide and Ithalond in the HPverse), and all omnipotent beings. These have all been known to pop in and out of the continua at will, entering badfic without portal technology. Notably, Sekhmet and Imhotep are both (non-omnipotent) deities who are part of World One mythology. Bloody Mary would be similar--as an urban legend, she is part of modern mythology. So in this case the agents would be dealing with the real Bloody Mary. Not being omnipotent (if she were, badfic would force character replacement rather than OOC), she'd be forced to go along with it until someone did a few exorcisms or cast a few Avada Kedavras... at which point, watch out.

1) I think they'd almost certainly be kept in FicPsych until they can be treated and restored to health or, the rules of time and aging permitting, they die. Maybe they'd be sent to a mental health care ward in the Real World, if possible (non-Humans certainly wouldn't go unnoticed), but I'd think they'd be kept in FicPsych because, presumably, with all the advanced technology, magic, and plotholes available, there's a much better chance of treating them there than elsewhere. I certainly don't think they'd be left to wander HQ; even PPC agents aren't so callous as to ignore a (potentially - scratch that - definitely dangerous) loony who's wandering around. IF they're found (and don't get lost in the plotholes of HQ corridors), that is.
However, I don't know if this question has ever been raised before, so don't take my word as final. Let's see what others say.

2) I'm not sure. Not being a specific character, there would be no way to measure a folk character for OOCness - and actually, in a Potterverse fic, something like that is more likely to be allowed than not, given the nature of their magic. Artifican urban legends, however, might be something you can treat more like a standard continuum. Still, what isn't written down in prose isn't a Word World, so chances are you'd have a mess on your hands regardless. Personally, I'd just avoid taking on badfics with those kinds of random elements.

1) I would say any of those. Depends on your liking - like so many things ;-) Seeing as the Flowers are not much concerned with the well-being of the agents they wouldn't take any particular action as long as you stay away from flamethrowers and such ;-)

2) You could just exorcise it away I guess (in the Bloody Mary case) or it would disappear as soon as the continuum fixes itself. I don't see any reason, why artificial urban legends would be any different.
There is one more idea - take them to the Supernatural continuum and let the Winchester brothers take care of them ;-)