Laugh at is usually used with joke (laugh at a joke-like here) or when someone is directly making fun of someone (laughing at someone) Laughing about is a bit more general, it just means finding something funny and laughing about it. So maybe not mocking them to their face, just laughing about them.

In the first sentence, "about" and "at" can be used almost interchangeably. The difference is that "at" singles out the person himself whereas "about" points instead at the fact of being thin, etc. In the second sentence, "about" would, in most cases, be a mistake. It is possible to laugh "about" a joke but it doesn't mean what you might think it means. If you laugh about somebody's joke, it means you are laughing about the fact that he told that joke, about the way people reacted to it, or the way he said the joke. Laughing "about" the joke does not indicate whether or not you think the joke itself is funny.