First, these extensions are not a necessary part of the anti-virus, and tend to cause more issues than they are worth (they duplicate functionality already in Firefox) and have no impact on the actual anti-virus protection (which always runs regardless of what browser you are or are not using). I strongly advise uninstalling them.

Second, it is the responsibility of Kapersky to ensure their extensions are compatible with Firefox, mozilla cannot ensure that all extensions are compatible with every version of Firefox.

Current Firefox releases require extensions to be signed to prevent abuse by malware. In Firefox 43 and later unsigned extensions will be disabled by default. You can set xpinstall.signatures.required to false to disable this security feature in the current Firefox releases. In later Firefox releases and Beta builds (47+) you may not be able to override this anymore.

See also:

• http://www.ghacks.net/2015/12/16/what-you-do-when-firefox-disables-installed-add-ons/
• https://support.mozilla.org/kb/add-on-signing-in-firefox

I don't know whether the version on Kaspersky's site is newer, but the latest I see on the download page is:

Kaspersky PURE 3.0 Latest Version: 13.0.2.558abcd Commercial Release date: 12 February 2013

Definitely those old extensions are obsolete.

How are you keeping the main program up-to-date with modern security threats?!

Oh, I missed this page: http://support.kaspersky.com/us/9636 -- I don't know when/whether you'll get new extensions for this product.