A proposal using suggestions in @GentlePurpleRain's answer, slightly modified. Texts are limited to 1000 characters (and I think that's before rendering) so I had to cut GPR's draft down a bit, especially to link the archive. I'm open to edits and/or changing the order of the items.
Be nice
Users are expected to be polite in chat, and swearing is generally frowned upon. This is something you need to be particularly careful about when discussing sensitive topics like politics, sex, and religion. If someone tells you to stop, stop.
Don't use chat as a 'replacement' for the main site
Though we'll obviously talk a lot about puzzling, full puzzles are better off on the main site. In chat, fewer people will see your puzzle, you have less ability to edit/update, and answers are disconnected from the question. Plus, chat Q&A don't come with "imaginary internet points"! However, feel free to use chat to share links to puzzles that regulars may find interesting.
Cryptic Clue Chat Chains (CCCC)
Since November 2016 we've had a "cryptic clue chat chain" (CCCC). A setter posts a archived.
Anyone can play, but please make sure you're familiar with how cryptic clues work (see the link above) before attempting to solve/post. Queries and collaboration are welcome. Post a solution by "responding" to the latest clue with an explanation. If the setter indicates that you are correct, you get to post the next clue.
Sometimes if the setter of the clue knows they will be away, they will post an MD5 hash of the solution. This can be used to verify a solution, but it is considered bad form to reverse-engineer the hash to obtain the answer.