Just to confirm, when I start the import process, my Classic board can stay open to users and run fine with no problems?
I wouldn't... Any new posts either won't come over or will not be linked to properly in threads.
Is it correct to say that the only real concern is that when I want to go live with the imported board, I should consider closing my Classic board, so that I don't lose any new posts users made during the import process?
well, when you go live, you'll want to close the old board, otheriwse people will be posting at the old site...
You use the redirect scripts to send users accessing the OLD forum location to the NEW one... You'd do this when your threads install goes live.