Is it possible to see inside the Blender user interface what the original frame rate of an imported movie (video file) is, so that one can quickly decide which frame rate to use for the rendered animation, without having to consult 3rd party software or non-cross-platform operating system/file browser functions?
No, Blender cannot currently expose the frame rate of Loaded movie media. Perhaps it is something that python scripting could access via the internal implementation of ffmpeg (the media decoder inside Blender).
When you import a movie into Blender there is an option on the import panel to "Use Movie Framerate".
This will change your current scene's frame rate to match the imported footage. Otherwise you will have to look at the mismatched length of video to its audio, then alter the scene frame rate until they match up.