Are you talking about text or image files? How are you downloading the files?

Image files have no file-name extension when I drag them to the desktop from the browser.

Start Firefox in Safe Mode {web link} by holding down the <Shift> (Mac=Options) key, and then starting Firefox.

A small dialog should appear. Click Start In Safe Mode (not Refresh). Is the problem still there?

Start your Computer in safe mode with network support. Then start Firefox. Try Safe websites. Is the problem still there?

http://encyclopedia2.thefreedictionary.com/Linux+Safe+Mode Starting Any Computer In Safe Mode; Free Online Encyclopedia

Yes, it is.

Maybe your issue is directly related to osX. Try this:

For one file

1. Select a file, then choose File > Get Info, or press Command-I.
2. Click the triangle next to Name & Extension to expand the section.
3. To show or hide the filename extension, select or deselect “Hide extension.”

For all files

1. Choose Finder > Preferences, then click Advanced.
2. Select or deselect “Show all filename extensions.”