The chats might be useful, this sometimes happens when people have an informal discussion. Perhaps it is good if the chats are always attached to a category of some kind, a tag? you want to have a generic chat about a subject, then attach the chat to the most specific tag you can find.
The easiest way to implement this is to make a heirarchical tag system, where every question gets a tag, every followup question gets a subtag, and related questions get a category tag, and so on. A question can have additional tags, but it should have a main place in the tree. Then the tree-structure of the tags follow the tree structure of the questions, with additional tags that represent domains of interest, subfields, and fields. Then there would be higher level tags, like "The KPZ equation", which contain all the questions related to this equation, and low level tags "How does Hairer's solution to KPZ equation resemble the heuristic analysis?" (lower level question, and also tag), and even lower level tags "How does equation 3.11 in Hairer's paper compatible with the analysis in KPZ 86?". Then a discussion can choose any one tag or tags from the tree to insert itself, this is the primary tag that locates the discussion, with additional secondary tags which are like soft-links.
This way, a person can look at all questions/submissions/answers/discussions under a given tag, and find all the relevant material. It is important for the site's future growth, without fine-grained compartmentalization, the site is bounded in size to what a person can memorize.
I know this is not so simple to implement, but it would be useful if the site is to be unlimited. Stackexchange is grossly limited in it's ability to scale up to thousands of users, it functions best when there are hundreds of users and dozens of questions daily, not tens of thousands of users, and hundreds of submissions daily. I think we should be focused on being capable of handling the heavier load, because, you know, if you build it, they will come.