When using Export to File, it appears that galaxy renames each output
file to something generic datXXX.dat and changes the filename
extension (to .dat). How do I convince Galaxy to preserve my
filenames so I know which file is which ?
There is currently no way to do this but it would definitely be a useful option to have. I've opened a ticket that you can follow and/or comment on if you're interested:
Is there some other way to download multiple output files and preserve
their names and file extensions?
Not currently, but this is on wish list and will likely get done at some point in the future.