dudduddud,
Notso Fantastic gave a good insight into what usually happens in the 'normal' world. Here's a response to your 3 questions from a slightly different perspective -
(1) What happens to the flight crew afterwards? - Routinely suspended without prejudice pending the outcome of any investigation to follow.
(2) Provided they weren't found to be at fault, do they usually fly again? - Yes, and to go a step further, even if found to be at fault, are usually able to resume duties after retraining / counselling etc. provided that no negligence, criminal intent etc. was determined to be the cause.
(3) Do flight crews often suffer from acute burnout after really off-the-wall emergencies sufficient to force them out of the job? - No most go back to work with little difficulty. It's part of one's own psychological conditioning that such things can happen, and we remain ready to cope with the unthinkable. I have a number of acquaintances who have been through horrific circumstances, terrorism, fatal crashes etc. and go back to work. Most just go home, have sex, most have a drink, I'd smoke 2 packets of cigarettes, and get up the next morning ready to do it all again.