I want to know whether there is any app on the Google Play Store which
- stores all dropbox files to memory card whenever there is net available
- Leaves the files in my memory card even after I uninstall the app
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You can do this by
clicking on the arrow on the extreme right of the items you want to save
cick export and save it to a selected place on your sdcard
For the 2nd point, you can copy or move the downloaded Dropbox files in another folder before uninstalling the application. Take your favorite file manager and do it.