I was a little bit worried coz in Firefox the Kaspersky addon are not enabled or something like that but on my Google Chrome this addon just worked fine. Do you think this is just ok even if it looks like it is not protecting my Firefox browser? (it is just my thought..) How to make the Kaspersky addon to works smoothly in firefox?
Generally Kaspersky is slow to update their firefox extensions so new firefox versions ( especially ones that have under the hood changes) bork KIS addons. Either you can downgrade your firefox or wait for an update from Kaspersky. Be sure to check that is the case though.
Are you using 2016? Patch D is in beta testing. That will make it compatible if it is not. Your malware protection should not be affected.