Yes you can bring both. Just remeber they check bags and you have to have backpacks and bags in front of you or beisde you so you dont hit something. (That can be kinda annoying)
For a locker deposit a quarter and get it back when you're done
I did not see anyone sketching when I was there. I did see several backpacks. I suggest if you are at the museum check at the front desk. Or call ahead to the administration office . I might add this museum is wonderful and will take a lot of time to enjoy.
Yes, I think you should be able to bring a sketch pad. Photos are allowed in most of the museum, so sketching should be fine. Backpacks are allowed, but you have to wear them on you front or hold them at your side. Lockers are available to store stuff, but I'm not up on details of size or cost for those.
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