There are two version of Flash, ActiveX for IE and Plugin for other browsers. You need to install the Plugin version for Firefox.

1.Download the Flash setup file from here:
Adobe Flash - Plugin version.
Save it to your Desktop.

2. Close Firefox using File > Exit
then check the Task Manager > Processes tab to make sure firefox.exe is closed,
{XP: Ctrl+Alt+Del, Vista: Shift+Ctrl+ESC = Processes tab}

3. Then run the Flash setup file from your Desktop.

• On Vista and Windows 7 you may need to run the plugin installer as Administrator by starting the installer via the right-click context menu if you do not get an UAC prompt to ask for permission to continue (i.e nothing seems to happen).
  

See this:
http://vistasupport.mvps.org/run_as_administrator.htm

Other things that need your attention:

Your above posted system details show outdated plugin(s) with known security and stability risks that you should update.

• Next Generation Java Plug-in 1.6.0_17 for Mozilla browsers

Update the Java plugin to the latest version.

• http://www.oracle.com/technetwork/java/javase/downloads/index.html (Java Platform: Download JRE)