1) I invited one of my users into the chat and the pop-up won't stop popping up on his system. It happens everytime he visits a forum.
No idea, what browser is he using?
2) Are there instructions somewhere about what all the stuff in the Chat CP does?
In the Chat CP Anything specifically? Seemed fairly self explanatory to me and Im a noob
2) Are there instructions somewhere about what all of the 20 buttons at the bottom of the chat window do?
They are basic IRC Commands that unless you own the IRC server serve no real purpose for you. Op and Voicing give users elevated status in the channel. (Voice allows for private conversation among the regualr chat if I recall, OP allows you to voice users, kick, ban etc.) Do a search for IRC commands for more indepth information. Very common and very wide spread, but nothing to fear
3) Will my non-moderator and non-admin users be seeing these buttons too? Is that appropriate, and if not, how do I remove them from member view?
No it checks for Admin/Mod status before drwing them, you can turn them off in the Chat CP, just have to stop the jumble of sentences from spinning around to actually spot it
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Anything specifically? Seemed fairly self explanatory to me and Im a noob 
