Can the display casing from a 13" MBP be used as a replacement on a 13" Aluminum Unibody? Can the display casing for a 13" MacBook Pro be used in lieu of the corresponding part for a 13" Aluminum Unibody MacBook??

Both screens have identical specifications and appear to have the internal connectors. Also, with the new aluminum construction heralding being a new design, it's doubtful that Apple made major changes to the frame or significantly modified the internals when they refreshed the old aluminum unibody MacBook.

That being said, I don't have access to either of these screens, and a fairly innocuous difference (say a difference of an inch on one of the internal cables) could render them incompatible. Another opinion or, better yet, first-hand experience, would be greatly appreciated.
No - the cable routing for the pro comes to the left side and the MacBook on the right. Check fixit.com for the details - it matters if you have a 2009/2010/2011 MacBook pro - each part is slightly different.