Firefox doesn't shut down cleanly. It never has. It often only does a "top kill" which leaves the bottom running. It won't start again until you kill the entire process. I have to open Task Manager, hunt around for the remnant of the application ("firefox.exe"), and kill it.
How can I get FF to shut down cleanly?
Sounds like it is hanging when you exit and later try to restart Firefox.
You need to update your plugins. It is important to keep them updated due to continuing security fixes and improvements in those plug-ins:
You shouldn't take article titles that strictly. You can't include everything in a title. Hanging means that the Firefox process doesn't end completely and that the process is still active and visible in the Task Manager on the processes tab.
You can try to disable the Windows Presentation Foundation plug-in (Tools > Add-ons > Plugins). If you have the Java Console and the Java Quick Starter extensions (Tools > Add-ons > Extensions) then you can try to disable those as well.
The only way you are going to determine what is causing that problem is to do a little troubleshooting, by trying the probable solutions offered in this forum. The suggestion that cor-el made was to disable those plugins and see which one is causing that "hang"; do them one at a time to help figure out which is causing that problem. Also, Firefox needs a restart for disabling some extensions, so waiting to disable them immediately before closing Firefox may not work.
How is it a "bad design" if you install a plugin that doesn't work properly with Firefox and that plugin causes the Firefox to hang and not close completely?
It strikes me as pure laziness to keep offering this same mickey-mouse solution to a chronic Firefox problem. Java, for example, is an integral part of Firefox and FF should be properly engineered and checked to see it's not choking on Java (or any other process). To say that I can always run FF without Java if my FF isn't working properly is simply laziness. If Task Manager can shut down FF, then FF can implement a similar, efficient shut-down. It's a bad design issue, not an end user issue. And worse, it's a design issue that's been bugging FF users for years because FF support keeps offering this same bogus remedy: "Uninstall all your plugins and try again." It's time to come up with something different.
None of us here at Mozilla Support are developers, we're all users like you, who volunteer our time to help other Firefox users. We can't fix Java and we can't fix Firefox, but at least with Mozilla we have a voice with the developers via Bugzilla.
Java is not an integral part of Firefox, it is a plugin made by a different company. The Java extensions that cor-el mentioned aren't even necessary for Java to work in Firefox; the Quick Starter is just that, it opens Java a little faster when it is called for - but IMO it is flawed and causes problems for some users. The Java Console extension is really only useful for people who need to troubleshoot Java problems, AFAIK. IMO, that extension doesn't belong in the Java Runtime Environment (as it used to be called), it belongs in a "developer package" instead.
I also was having this problem and after searching for a week I came across a posting that recommended downloading SUPERANTISPYWARE, it is a free download and as strange as it seems this WORKED!! I could not get FireFox or Google Chrome to end no matter what I tried, I could not even use the Task Manager to close them and when I tired of waiting and tried to shutdown the computer it would hang waiting for the programs to allow a shutdown and I would finally give up and shut of the power,
I hope this works for others also.