I have rooted my Android 5.0.2 (Samsung Tab 8.4 Pro, TM700) the day before yesterday, and I noticed yesterday that my Chrome won't save files anymore, at all. I then installed Firefox, same behaviour. WhatsApp, can't download images or videos. My mail application downloads files just fine, so I am not sure what is going on here. What do I need to check for? Could it be permissions on some folder, and if so, what folder to check?
Try this in a Terminal Emulator. The 'su' command is for becoming the root user and the next one fixes all the permissions.
su restorecon -v -R /data/media