I can't click on a tab to select it or move it. I have to use the tab list and group to do this.
Note that Nightly is unstable, if you havent already noticed. You can try safe mode as a good first step to see if addons are the problem.
You should always have a Firefox release version as a fallback installed, so you can switch to that version.
Also make sure to use each Firefox version with its own profile (i.e. do not use the Nightly build with the same profile as other installed versions).
With the Australis code landing on mozilla-central anything can go wrong and you should be prepared for issues.
Best is always to first try a new test profile.
See "Creating a profile":
I noticed you have the Tab Mix Plus extension installed. You might try testing with Tab Mix Plus disabled in case it is incompatible with a change in Firefox 28.
Thanks all, it was Tab Mix Plus. After posting the question, I realized that I hadn't tried disabling add-ons, and when I did, Tab Mix Plus was the culprit. I still have Firefox installed, but I didn't know about the separate profiles, an omission which I will remedy. I needed Nightly, because I'm running 64 bit Java for Minecraft. Thanks for the help.
I just updated Tab Mix Plus to version 0.4.1.3.1 and it works with Firefox 29.0.