detecting and converting incompatible media files

You will still be able to use other software for playback or conversions when upgrading to macOS Catalina 10.15. People will still be able to play or convert on pre-macOS Catalina systems.

If a file opens with QuickTime X without converting, is that a good sign that it will still play after Mojave?

Almost certainly, yes. The issue is about abandoning QuickTime 7, and the formats and codecs that were used when that was current (2010 or earlier).

The Apple support mentions MPEG-4 remaining compatible, but what about files with .mpg extensions? What about MPEG-1 files?

See previous. If a file opens with QuickTime X without converting, that will probably remain true in the upcoming OS.

Is there any easy way to tell if a video file is 32-bit versus 64-bit?

32-bit versus 64-bit compatibility is about applications, not about documents.

Does its icon showing its thumbnail (versus as a generic icon) in icon view in finder help determine which files need converting and which ones will continue to play in QuickTime after Mojave?

No, I think not in a reliable way. E.g. AVIs may show correct thumbnail/preview poster image, but not play in QuickTime X without converting. QuickTime X may still be able to convert (some of) such files in macOS Catalina 10.15, though.