Copy Thread
 

Can you make a complete copy of a thread and move it to a private board for discussions like a Mod section?

You may go to Thread Tools > Copy and make a copy of the thread and put it in the mod forum. It won't automatically update when people add more posts to the original thread, if that is what you are asking about.

And is that the same settings for version 3?

What do you mean same settings? There are no settings for it - you just choose Thread Tools > Copy

Is it possible to make a copied thread that automatically updates on both forums?

I'm thinking of making a "featured discussion" forum, so I'd like to copy a thread over there to get it more attention, but leave it in its original forum as well.

There is no default option in vbulletin to do this. You would need to write a plugin that would create that post in both threads.