It is possible that there is a problem with the file(s) that store the extensions registry.

Delete the files extensions.* (e.g. extensions.sqlite, extensions.ini, extensions.cache, extensions.rdf) and compatibility.ini in the Firefox profile folder to reset the extensions registry.
New files will be created when required.

See "Corrupt extension files":

• http://kb.mozillazine.org/Unable_to_install_themes_or_extensions
• https://support.mozilla.com/kb/Unable+to+install+add-ons

If you see disabled, not compatible, extensions in "Tools > Add-ons > Extensions" then click the Tools button at the left side of the Search Bar (or click the "Find Updates" button in older Firefox versions) to do a compatibility check or see if there is a compatibility update available.

I have deleted those files and installed some add-ons to test.

I closed firefox but the process still continued and I forcibly ended the firefox from the task manager. I then opened firefox but my add-ons did not work.

Now I again closed firefox and waited for the process to end itself so that firefox has a clean exit. I then opened firefox and then I am

SURPRISED

All of my add-ons were working and all the previous settings of the add-ons were restored like Adblock Plus subscriptions and others.

but IDM did not integrate properly it is downloading each and every audio-video file on the webpage!