Are .apple files potentially synonymous (or a wrapper to) .dmg files?
The .apple files are probably either AppleSingle or MacBinary files.  The Apple file system keeps more information about a file than other filesystems (Window, Unix, etc.) do, so AppleSingle and MacBinary formats were created to package up all that information in a way that could be transported on any file system.  

So if you're looking for MyDisk.dmg but instead find MyDisk.apple, that's probably the encoded version of the file you want and you need to decode it.  Try downloading the free StuffIt Expander and see if it can convert the files for you.