Ok, so I'll bite first. Before I move into the answers, I just want to state that the opinions stated below are my own. Within the confines of the ToC (Terms of Conduct) and other governing moderator guidelines, moderators are people, they have their own views and work in their own ways. The answers below are as I see them and don't neccessarily match what other moderators may think.
- Does chat need to stick to workplace issues exclusively?
No. The cooler has been entertaining people for many years without having to stick only to workplace matters. Obviously, however, if anyone has a workplace issue that they need to ask, then the current participants drop tools and help as much as they are able, and the off-topic chatter usually pauses. I've seen this in action many times, and many people have been helped.
- What is off topic for chat?
A broad subject this one. Basically, subjects that don't conform to the ToC. Basically, "be nice" is over-riding, as is basic common sense and decency toward others... Trash-talking other people, moderator actions, and other sites obviosuly brings some attention and crosses the "be nice" line.
- What is grounds for a kickout from chat?
In short, disruption and obvious bad feelings being aired, profanity being aimed at others, and badness of that order. Also, reacting to baiting (lowering yourself to language you wouldn't normally use).
Be aware that people don't necessarily get kicked out of the room because they're found at fault - it's sometimes a matter of removing tension from a room by freezing out the loudest participants and allowing them the time to cool off, regardless of any empirical idea of "fault".
- Do mods need to warn a user before a kickout?
If there's a chance to, yes. However, moderators don't monitor chatrooms constantly, so sometimes it's a case of dealing with fallout as it's happening. Throwing people out or freezing a room gives people a chance to cool down, and the moderator time to read the transcript, try and work out what's gone on, and act accordingly.
- When is a room cooldown preferable to a cross-chat kickout?
Simply, numbers. If most of the chatroom is involved, then a freeze is a way of temporarily shutting up everyone at once.
- Is humor off topic in chat?
No, definitely not. But remember that it needs to be open and understandable to everyone. Sarcasm/satire can come across as abusive to people who aren't in the circle of trust. Remember that anyone can read chat transcripts, and anyone can visit and follow a chatroom without being logged in. Chatrooms aren't private.
- IF humor is not off topic in chat, what are the guidelines?
See answer above.
- How can mods avoid the appearance of favoritism towards or bias against users?
We can't completely avoid unconscious bias, it's part of being human. Also see below.
- What kind of deference should mods show to regular mods of a group when they are in chatrooms they do not normally visit?
Notionally, off-site moderators should defer to on-site moderators because there's more of an idea of context and history.
However (and this is a big point), the currently active moderators on the Workplace are regulars in chat. This presents a problem of unconscious bias, and being too close to the issues at hand. We rely on the judgement of off-site moderators to be impartial. As you're aware chatrooms are moderated by any sitewide moderators - they react to flags or calls for impartial assistance. You'll see them sometimes pop into the chatroom, and most of the time they'll leave again because there's nothing that requires special attention. It's not because they're spying or selling popcorn.
- What steps are being taken to avoid the appearance of impropriety?
We talk about stuff. We argue, we listen to each other. We're not perfect. We attempt to be impartial, but see above regarding unconscious bias.
- How should people with social difficulties, such as autism, severe introvertedness, or non-native speakers who may use the wrong word, be treated?
No differently to anyone else. We are used to all of the above and we're open to different ways of expressing things. However, intent is intent and is usually pretty clear.
Through all of this, be aware that the cooler has been largely unmoderated for a long time. Regulars have got used to a particular way of chatting, and a lot has been allowed to happen in the past.
Note that entering into discussions about particular viewpoints or events isn't likely to be productive.