This is an issue I sometimes had in OS 8.5. Double-clicking would result in LW opening. It has something to do with LW being seen as the creator of the file. The solutions I can think of are: 1. Rebuild your desktop (Good healthy thing to do anyway, even if it doesn't help your issue), 2. Drag & Drop the file onto the application that would normally open it. (also the option of resaving it.), 3. Use a file utility like Mac Army Knife to adjust the file's attributes so that it is then always opened by the appropriate app., 4. Go to your File Exchange control panel & set the appropriate apps to open these files by giving examples of the ones you're having problems with.

Personally I mostly had an issue with .sit files. I either use Mac Army Knife to solve the issue, or drag & drop the sit files & then recompress them for archiving.

As for mp3, some mp3 players will automatically change the creator when you open the mp3 with them in OS 9. There's a free iTunes plug-in/app called 'iTuner' that will change the mp3's icon so that it always opens with iTunes. You can find it at most mac shareware sites such as c/net download.com

mpg's, I'm not so sure of an easy fix for unfortunately, other than points 3 & 4 above for a permanent fix.

Which version of LW are you using? I should imagine/hope the later versions for OS 9 would have overcome this issue. Get back to us if you're still having problems.
The suggestions I made do work. Mac Army Knife is a shareware, but just be careful how you use it. Compare the file's attributes to another file of same type that doesn't have a problem & adjust the details accordingly. Generally only need to change the creator, perhaps type, & perhaps the version but that's optional. (changing the file's attributes other than the ones I've mentioned may damage the file - so backup just in case if you're not sure!)