Firefox has three separate "release channels" that run in parallel: stable release, beta, and aurora. The update feature does not cross between the three channels, but you can install the version you want directly from the channels page on mozilla.org. Naturally, a backup is wise before installing a cutting edge test version. This article discusses some options: Back up and restore information in Firefox profiles.

On the Java question, if Firefox was running during the installation, you might need to exit and restart Firefox before your new plugin is detected. Then it should appear here:

orange Firefox button or classic Tools menu > Add-ons > Plugins category

If it appears disabled and can't be re-enabled, perhaps the update didn't work well. Which version does Firefox show?

Firefox release (currently 17.0.1) is the most stable and reliable version of Firefox.

Firefox Beta (currently 18) is the test version of Firefox. Bugs are ironed out and features are tested before they move to release. This is slightly less stable than release, but has newer features.

Firefox Aurora (Currently 19) is the version before nightly, where some new features are landed before moving to beta. This is fairly stable, but you may run into issues on Aurora that don't exist in more stable build. at the same time this has more new features than beta.

If you are just wanting to use Firefox for everyday use and aren't a web developer.Mozilla contributor or tester, I'd recommend using release or Beta.

for the Java issue, there is a new version of java, 7 update 10, which java.com doesn't recommend yet. You should still update from http://download.oracle.com/otn-pub/java/jdk/7u10-b18/jre-7u10-windows-i586.exe.

In other words Aurora is not a version of Firefox nor a upgrade but rather the name of one of the development channels where it is 19.0a2 currently.