Being taken to the woodshed is obviously a reference to be taken out back to a private setting for a spanking.
That preparation goes something like this:
"You have to listen to the question and answer it truthfully, but don't offer up more information than you need to. He'll be trying to get all the information he can from you, and your job is to give as little information as you can without outright lying. For example, if he asks you if you can tell him what time it is, how do you respond?" ...
"Wrong! the correct answer is 'Yes.' That's it! Yes, you can technically tell him what time it is. Because you have a watch! But you shouldn't volunteer that it's 12:15 pm, much less talk about how fucking cool or expensive your watch is!"
I'm not making this up. It's an actual term. See. Told ya so. The 5th Circuit Court of Appeals (the one that covers TX, LA, and MS, natch) actually opined on witness preparation.
Whether those that use the term think about the spanking related etymology- I don't know, but I doubt it.
Bring the Wood:
It's a real phrase. In football it means a punishing hit. Seriously. It's been adopted by some of the nation's best teams as a mantra. Seriously.
I sincerely doubt that users of either of theses phrases consider that they originated in corporal punishment, but every time I hear one of these phrases I think of spanking, and someone getting butthurt.