Lunar, CRM is "Cockpit (or Crew) Resource Management".
It has nothing to do with whether an FO justifiably or otherwise has a word with management about a Captain.
It is nothing to do with everyone on a flight deck having a nice, warm feeling about the person sitting next to them.
It is use of all resources available to you to complete a flight safely, legally and expeditiously.
Yes, it is taught. There is an initial course and then you are required to have annual refresher training.
As far as personality differences on the flight deck are concerned, it happens, as it happens in all other walks of life. Professionalism, however, dictates that you don't let that get in the way of a safe flight. You save arguments until you're on the ground. Thereafter, if you and a skipper want to go round the back of the building and resolve your differences, do so.
Similarly, personal problems can weigh heavily on a pilot's mind. A new baby, or divorce, or bereavement, or money troubles can all distract. Whilst these would not usually result in a pilot being unfit to fly, it is wise to give the other guy a "heads up" to let him know you may be a little distracted and to keep an eye on you. I did this with FO's I flew with just after my mother died. It's not a big deal. If it is, then there is nothing to stop a pilot requesting a bit of leave. If they know the circumstances, most Fleet Managers would be completely happy to grant it.
And finally, as far as discussing pilots' actions publicly, there is a fine line to be drawn. There's only one person who knows where it should be drawn, and he forgot. There's an old adage - "Learn from others' mistakes, because you won't have time to make them all yourself". Aviation works very much by sharing information, learning from other incidents and putting in place systems to prevent an incident that occurred in another airline recurring in yours. It's called being proactive. However, "publicly" is taking it a little far. Nobody within aviation that I know, for example, would be happy about CVR or Cockpit Video recordings of incidents being played on public radio or TV stations. The public have a right to know how an accident happened. They do not have the right to hear all the gruesome details of a pilot's last few minutes of life, nor do they have the right to those tapes to use in any lawsuit against his relatives and estate. Nor, further, should they be used in any criminal action against flightcrew.
Hope this all answers your questions.