Hmm, in the Question Details > More System Details next to your post, the plugin enumerator reported:

Next Generation Java Plug-in 11.40.2 for Mozilla browsers

So there appears to be a problem with either multiple plugin files or corruption in the settings file that stores plugin data.

As a Windows user, I don't know all the details of where to look for this problem on a Linux system, but one place to check would be here:

Could you return to the about:plugins page and use Find (Ctrl+f) in the page to look for npjp and see whether you have both the current plugin and the older version on different paths.

In the case of duplicates, you might need to physically remove the older library, or if another application requires it in that location, perhaps it can be replaced with the newer one. Or maybe Firefox is finding it though a link that is not needed.

Anything along those lines?

Thanks for the quick response! I used to have multiple Java plugins getting picked up by Firefox but I have remedied that and restarted (and had done so before my original post). There is currently only one version shown in about:plugins.

The "11.40.2" is the plug-in version - it is versioned independently of the JDK with which it ships. For example, in JDK 7 the plug-in version was something like 10.x. Here is a copy/paste of the only libnpjp listed in about:plugins:

   File: libnpjp2.so
   Path: /java/jdk1.8.0_51/jre/lib/amd64/libnpjp2.so
   Version: 11.40.2
   State: Enabled (STATE_VULNERABLE_UPDATE_AVAILABLE)
   Next Generation Java Plug-in 11.40.2 for Mozilla browsers

I have also ensured that the version of 'java' that shows up first in $PATH matches the JDK this plugin came from. I've also tried setting JAVA_HOME before invoking Firefox (I've seen some StackOverflow posts suggest this). No luck...

While the location of info on Linux is different than Windows, I still think the 40 should be a 51, as in:

File: npjp2.dll Path: C:\Program Files (x86)\Java\jre1.8.0_51\bin\plugin2\npjp2.dll Version: 11.51.2.16 State: Enabled Next Generation Java Plug-in 11.51.2 for Mozilla browsers

If Oracle didn't update the internal version number in the file for Linux, hmm, that's going to be an ongoing problem.

It's possible that the .so file has the correct version but Firefox's pluginreg.dat file is corrupted and not being updated to match. Please see this post for suggestions in that case: https://support.mozilla.org/questions/1049282#answer-703069

Thank you! I'm making progress. I wasn't aware of the correlation between the minor version in the plugin and the update number of the JDK. I closed Firefox, deleted pluginreg.dat, restarted Firefox, and the versions now match what you show. Furthermore, the JDK is no longer marked as vulnerable, only enabled, and it now runs without asking for activation. So that resolves some of the weirdness...

However, the Java version applet still does not appear to run. There is empty space, although the <applet> shows up in the DOM. Eventually it disappears from the DOM and is replaced by text saying it was unable to determine my version. Any other ideas on what I could check to see what's going on with the applet?

I've run Firefox from the terminal and the following output is printed every time I click on the applet:

[4044] WARNING: pipe error (70): Connection reset by peer: file /builddir/build/BUILD/firefox-36.0/mozilla-release/ipc/chromium/src/chrome

The first number in parentheses varies: after 3 clicks I see '68' twice, '60' a bunch of times, '78' once, '70' 36 times. Gonna dig into what that may mean...

Hmm, I really don't know how to interpret that, sorry.