Could you add some info on what each tag does?
Oh yeah, sorry about that! I'll add an 'Output' and 'Description' column in the table.
I'm also interested how you found these. Lurking over SMF code? I had no idea.most of these even existed

I'm actually working on a detailed BBCode documentation for a while now
(something that is different from the existing guides of course), since some of these codes aren't included in the
SMF Bulletin Board Code. But I'm still not ready to publish it yet, I'd say it's around 80% complete.
It seems to be useful. I just can’t understand the purpose of "No BBCode" button.
No BBCode prevents the tags from being parsed, that's why I was able to write [nbsp] in the OP.
Another concern I have is that the new icons rely heavily on the talkimg service. What happens if that service stops working?
The buttons should still work but will show the alternative text instead. Unfortunately, that's one limitation of the script, unless the icons get added to the forum's official resources.