It depends entirely on the state you're in, the mood of the person he stole the card number, and of the prosecutor if one has gotten involved. You can't get an answer to this question without telling us more. But your son, although he is a minor, could be in some trouble.
Almost certainly not. Being offended by someone's inappropriate or insensitive comment is not usually enough reason to bring the legal system into action against him. Rudeness is not against the law. But you may find satisfaction by complaining to the company that owns the site (over the boss's head) or others that are connected or affiliated with that business.
There is almost certainly no benefit to not picking up the letter. It will only make it appear as if you are evading communications about the debt. Also, if you did not get any other copy of the letter besides the certified one, you could really prejudice your ability to defend against the debt -- or to settle the dispute, perhaps -- by not being aware of what is in that letter.
The photograph has nothing to do with you, unless you are saying that it is part of the disparaging posting in the sense that someone is falsely and claiming that it is you. Even then, if it is obviously not you (say it is a picture of a monkey or a famous person) the photograph could not be defamatory, but merely a (mean) expression of opinion. It is as if the person who posted is saying that you are "just like" the picture. That is a terrible thing to do, but in the United States it is...
Your first step is to dispute the charge with your credit card company. The burden of proof on their claim that you took the equipment is on the hotel, not on you. You may very well need to get a lawyer (in California) to help you out here and "remind" the hotel of this fact by a letter written on his stationery.
If you have proof of delivery you must submit it in writing to the charge card company. That is almost always enough to at least move the ball back to the other side. Is the credit card company telling you this information is not sufficient? If so, why?