I had no trouble viewing the Flash at the link you gave.
If you are getting no SWF file type listed in FF where I asked you to look, then you may not have the Firefox plug-in installed properly even though about:plugins reported that you have it. Will have to check as to whether it can be added to the Manage... window in FF, but if plugin was there to use, it should be listed. You could check the Plugin folder for Firefox to see if it's there.
P.S. You're not using anything like AdBlock or FlashBlock are you?
The Adobe Flash Player Plugin is the one you need. Might try to re-install in from the link provided.
OK, I installed the plug-in from the link you gave and it makes no difference:
* about:plugins still says that I have application/x-shockwave-flash (Adobe Flash movie, swf) and that it is enabled
* SWF is still not listed in Manage...
* I still get the dialog asking me what I want to do with an swf file when I go to a site that uses Flash
* I still have the file association of SWF files to Firefox
How can I disable the last of these? Isn't it the case that once you've got an association of a file extension with an application, it always has to be associated to some application or other?