Looks like I get to answer my own questions...
Answer to question 1:
The missing entry in the Info.plist to solve the "cannot open files of this type" error message is the CFBundleTypeOSTypes entry. I had the others, but not this one. Here's a modified version of my Info.plist with context:
Answer to questions 2 & 3:
Adding the following handler allowed me to get the file that was being sent via "Open with".
on application_openFiles_(sender, droppedItem)
--- droppedItem contains the path to the file
I stumbled across a really old reference to it and decided to give it a try. It worked! It would have been nice if Apple's documentation had something for application_openFiles_. I would have expected to possibly see it here along with other methods the AppleScript App template inserts by default:
If there's a more recent handler I should be using instead, please provide.
Your link is the current documentation, and includes that method. The template just has the basics; you are expected to add other delegate methods and configure the Info.plist as desired. Note that you can set up your own project and file templates, although information about that is difficult to find.