The file xpcom_core.dll was last used in Firefox 2.0.0.x versions and is no longer present and used in current Firefox versions.
You seem to have a program or an add-on that is still looking for that file.
You will have to uninstall that program.

See Troubleshoot issues with plugins like Flash or Java to fix common Firefox problems
See Troubleshoot extensions, themes and hardware acceleration issues to solve common Firefox problems

I have the problem, even in safe mode, so it does not seem to be related to an extension or plugin. This error first appeared with the most recent update to Firefox.

I Googled xpcom_core.dll, resulting in the following link ... http://www.dll-error-fixes.com/fix-xpcomcoredll-error/ ), which indicated that this DLL is a Firefox plug in, and that it's apparently a Firefox-only problem. I found the file npmozax.dll file in the C:\Program Files\Mozilla Firefox\plugins\ directory, deleted it, and my "can't find xpcom_core.dll" problem appears to be gone in less than 5 minutes!

FYI: In Firefox 3.6 and older versions only extensions are disabled if you start in Safe mode, so it can still be a plugin problem.

I've also seen references that the Mozilla ActiveX plugin is connected with that issue.
The plugins list of the original poster shows that plugin.

The Mozilla ActiveX plugin is not compatible with Firefox 2 and 3 and you should remove it.

For instructions on how to remove the Mozilla ActiveX plugin see: http://kb.mozillazine.org/ActiveX

I had this problem too. But was able to fix it.

It seems that the xpcom.dll file in the Firefox program directory had become corrupted. I couldn't remove it, even after an uninstall.

After the uninstall the Program Files/Mozilla Firefox directory was still present containing the corrupt .dll.

Without renaming the corrupt directort a new install will just go into the old corrput directory and the .dll will remain corrupt...

I just renamed the directory to something like Program Files/Corrupt Mozilla Firefox and reinstalled Firefox in a different new directory.... problem solved.

Hope this helps you too.