Okay, I will answer this as a pilot, since that is an area of expertise for me.
First of all, how people react under serious stress has been recorded, and it varies with the individual. You can train people, however, when "The Real Thing" happens, there is always the element of unknown reactions. Although, the goal of all the repetitive emergency procedures training and exposure to situations in a simulator, is meant to reduce the likelihood of someone freezing up or reacting incorrectly.
As to, did they "merely" do their job question. I have always said that having a boring flying career is the most preferable state of events. The best pilots use their knowledge and judgment so that they don't have to use their skill. It is indeed a very ingrained part of pilot training to handle these emergencies. The amount of training is reflected in the significant investment it takes to become an ATP pilot, as well as the experience one needs to show in order to become a captain on one of the "majors". Most people have no idea how tightly regulated and trained pilots are, hence they may attribute some aspects of job performance to superior heroism, when in most cases it is training.
Keep in mind, unlike most career fields out there, pilots are annually subjected to the following:
- Practical Flying Exam
- Flying Rules Written Exam
- Aircraft Systems Written exam
- Weather written exam
- Emergency Procedures exam
- Aviation knowledge oral exam
- Full Class I Physical
And then, depending on the airline, or aviation field, they could also get psychological exams, additional testing, and of course, continuing education such as Crew Resource Management training, additional upgrade training and type ratings, etc. Being a pilot is unlike any other career field that I am aware of.
So, that's probably why you see such humility from pilots. They feel like they really are only doing their jobs, but the average citizen has no idea what they have done to become a pilot, so anything they do seems heroic. Although, that still doesn't change the fact that this was indeed a very special event pulled off by the crew (although not totally unique as Suma points out in his answer).
So, let's address each of your bolded questions now that I have given you an extensive pre-amble:
Are there studies of how flight crews react in emergency situations?
Every single accident is tracked by the NTSB. Part of the charter of the NTSB, FAA, and nearly any worthwhile flying organization is to use those accidents as lessons learned. In USAF pilot training, we have a book called "Road to Wings" which the students affectionately call the "Bloody Road to Wings" because it is a summation of all pilot training accidents, and what was learned from them. These lessons learned are also taken into classes such as crew resource management (which by the way is also finding it's way into the medical field because the principles are so successful).
Do they successfully follow their training and make appropriate decisions that maximize their chances of survival?
Not really sure exactly what you are asking here. All I can say is emphatically YES! That's the whole purpose of the training they receive. One can also argue that the reason that airline captains are paid so much (aside from the extreme training, and awesome responsibility, but you also want the guy to have something to live for when things go wrong (although I personally find that a poor argument)). Keep in mind, there is a history that got us to the point of having these checklists (like I mentioned, the Bloody Road to Wings). There is a manual that the USAF has for each aircraft that we call the Dash One. In it are things called Notes, Cautions, and Warnings. Each and every note, caution, and warning is written in someone's blood. Same way with checklists (and again, as I mentioned earlier, checklists are finding their way into medicine and other consequence driven fields).
Do they tend to panic and make mistakes?
Again, that's the purpose of the training. Keep in mind, things don't happen that often. Most of the time, when something does happen, it doesn't even make the news because very few people even know something is happening (for instance, a hydraulic failure, or something like that, is generally never broadcast to the passengers unless it's incredibly severe, and they need to prepare the passengers). Overall, when we DO have accidents, it is a result of an error that led the crew into an unrecoverable situation (what we call the "error chain"). The dual flame out bird strike is not really one of those type of events (like flying into icing conditions with anti-ice off like happened near Buffalo, NY). At that point, the pilot hasn't been part of a series of mistakes that are building up, and thus can hopefully let all that training and preparation take over.
Are there any meaningful figures on this subject?
I know that the USAF keeps a lot of figures and statistics from all aircraft investigations. They also have a very formalized flight safety program that has publications, courses, and officers in each and every flight squadron whose job it is to promote the training that will let pilots do their jobs when the feces impacts the rotating air oscillator.
Sorry, was sort of free flow typing there. let me see if I can bring some other sources into the discussion.