Can military judges be dishonorably discharged for dereliction of duty?

No they cannot. Failure to act (even if that were shown to be true) is not in itself grounds for a dishonorable discharge.

In fact the ruling from the judge in question is completely right, although unfortunate. The President, as commander in chief, should NEVER have commented on such matters. It creates a situation where the accused can no longer receive a fair trial and sentencing. In law, the judge acted correctly. It is far more important that the law (even military law) be SEEN as fair to all than it is for a particular felon to be imprisoned under a cloud.

Get mad at the President, not the judge.

And no, the President cannot be impeached for being a dumbass.

"Can, therefore, the judges be dishonorably discharged for not protecting the soldiers, sailors, and airmen/women under their protection?"

Only if we can the President impeached and imprisoned for his failure to do the same, since legally, he has more responsibility for protect airmen/women than the judges do. The judge is there to ensure the trial is fair, not protect people.

And it actually does take a genius to be able to sort through all the comments being made on Sexual Assault right now, and then to ignore them in sentencing guidelines. Here's an entire page worth of really smart people (all JAGs) discussing this problem and what it all means)

http://www.caaflog.com/2013/06/14/what-might-be-a-...

You may think it's just referring to cover-ups, but it's worth noting that wasn't how it was reported in the news at the time (which isn't to say the news is right, but the message they put it out does influence people's perceptions, which is what the judge is trying to prevent.