What is the best use of chat vs when should you post or comment?
A rule of thumb (which I just made up) is, if what you have to say is oriented toward the site -- for example, you're pointing out a logical or factual error in a post, or asking for clarification -- then it should stay on the site, i.e. as a comment. If what you have to say is oriented toward yourself -- you have a minor question, perhaps prompted by a particular Q&A, but not actually about it -- then chat might be a good place for it.
Is it 'anything goes' conversation?
The same "be nice" rules apply to chat as to the rest of Stack Exchange. Avoid profanity, and never use ridicule/put-downs, not even in jest. You don't have to stay on topic, though it's nice if you can manage it. However, if you wouldn't bring up a topic as dinner table conversation at the office holiday party, it probably doesn't belong in chat, either. (Assuming your office isn't a fan of knock-down-drag-out dinner table fights, that is.)
Can I ask people to proofread a sentence or two?
Sure. Be patient, and be prepared to accept "sorry, no time right now" as the response. (Or, more likely, be prepared to be ignored if nobody is in a proofreading mood when you post.)
I just can't remember this word on the top of my tongue! What is it?!!
I presume you meant it's on the tip of your tongue. :) You can certainly ask in chat, especially if you don't have (or can't be bothered to provide) the details that a single-word-request question would require. Whether you get an answer will depend on who else is in the room with you, and how well their brain synapses happen to be firing.
Can I ask more opinionated questions that just would not go over on main? I've tried that but got little response.
You can ask. No guarantees about getting an answer; as mentioned, with chat, it all depends on who is in the room with you.
I can't get a word in edgewise ... things go by so fast
Chat is never a linear conversation. If you have a comment to make about something someone said 5 minutes ago, go ahead and post it, even if the conversation seems to have moved to an entirely different topic (or three) since then. If you make your comment an actual reply (by clicking the little angled arrow pointing to the right that appears when you hover over the 5-minute-old post), people can simply follow the link to figure out what you're talking about.
It does help to be able to type quickly. But another skill is just as valuable, and much easier to learn: do not worry too much about typos. Unlike many other chat systems, SE chat allows you to edit your posts (for a limited time interval, obviously), so if you fat-finger something in an effort to post it in a timely manner, you can go back and fix it. Or, if people seem to have understood what you meant, don't even bother with the editing. (Yes, that's easier said than done, at least for a certain segment of the ELU population.)