Terry, again sorry for the delay, be away from the machine...
No it's OK. The second line is the grep command itself.

Duh, it is there, right in front of my nose, and I miss it!

Before continuing, I should mention - thought I did, but alas - that I am using KDE!
If you only update via the repo there shouldn't be an issue but what I meant was to run the "ps -ef | grep firefox" at each stage of the steps you outlined before. So:
[...]
Thank you for taking the time to try to help me out, much appreciated. I have followed your suggested steps, with the command in between each and every one of them. There are no surprises, and only one instance all the way... except when FF isn't running, and there is no instance showing.
When starting - first time - from TB:
cj 3169 1 15 23:50 ? 00:00:00 /usr/lib/firefox-6.0.2/firefox-bin http://www.pclinuxos.com/forum/index.php?topic=96708.new;topicseen#newWhen TB links are non-responsive and FF is started from e.g. desktop icon:
cj 3413 2518 19 23:51 ? 00:00:01 /usr/lib/firefox-6.0.2/firefox-binIn between, it's just the grep line alone, i.e. no other instance of FF running.
Also what was the result of running:
gconftool-2 -g /desktop/gnome/url-handlers/http/command
gconftool-2 -g /desktop/gnome/url-handlers/http/command
/usr/lib/firefox-6.0.2/firefox %sI'm hoping this will give me a clue as to what is happening.
Hope it will, as I am clueless...

Thanks again!
I encountered the same problem with Thunderbird-Firefox. What I did to turn off Firefox's nagging is to go into the Firefox preferences, Advanced section, General tab. Uncheck the "Always check to see if Firefox is the default ...". Then click the Check Now button. Answer the prompt, if there is one. Firefox should stop nagging for the default browser check. When Firefox is updated, you will need to click the Check Now button again, but only once.
DJohnston, thanks for the suggestion. I was reluctant at first, as I wanted to understand
why this happens - and have it fixed - but the pester factor won over and I did what you suggested... it actually turned out worse, in some sense: The same behaviour with TB persisted; on top of that, when closing one of multiple tabs (using CTRL-W), it started closing down FF itself (i.e. with all open tabs).
Why this should happen, I have no idea - It simply does not make sense! Anyway, for this reason, I reverted back to having the check enabled.
Thanks anyway, it was worth a shot!
****
Wondering... maybe I should try that old chestnut of either deleting the firefox folder in home; or even try creating another user... just to confirm if the behaviour is global.
Shall post this, then try and report back.
Cheers!
CJ