Is there any way to fix the annoying message that you sometimes get when clicking a link
Firefox is already running, but is not responding. To use Firefox, you must first close the existing Firefox process, restart your device, or use a different profile.
I must say I cannot reproduce it here...
$ apt policy firefox
firefox:
Installed: 96.0.1-1
Candidate: 96.0.1-1
Version table:
*** 96.0.1-1 500
500 http://httpredir.debian.org/debian sid/main amd64 Packages
100 /var/lib/dpkg/status
Did you try removing (after backing up) your configuration? Or to simply start firefox with a new profile?
You could also try firefox straight from mozilla.org to see if it makes any difference.
Also, when it happens, do you see firefox running in the background? (With htop you can press F4 and search for a firefox process.) If so, you can kill it, if not, there should be a leftover lock file. In that case maybe it is not closing properly. Do you explicitly close firefox (with Ctrl + Q) or you just logout? (I actually do not have this problem when I just logout, but it is better to close it before logging out.)
Do you have many extensions installed? If you have the patience, you can disable them and slowly enable them one by one and see if you can find if one them is causing the trouble. (For the record, I have uBlock Origin, Disconnect, Privacy Badger, Facebook Container, and Plasma Integration (plus a couple more) running without problems.)
Just a few suggestions...