please be more careful when providing the information. when people are trying to help you, it's difficult to find the solution if you provide the wrong info.
you first wrote:
you then posted:
these statements clearly contradict each other. then you also you the instructions for the firefox.desktop file as the actual firefox.desktop file you were using. obviously things don't work out if people post the instructions they were supposed to follow instead of posting the actions they performed.
when trying to obtain help, post your own configurations, findings, output, etc. as we all can get the instructions of the internet.

anyways, i'm happy you found the problem.

the reason why your firefox doesn't start up is because you created the wrong symlink.
you created a symlink to /usr/lib/firefox/firefox (which most likely does not exist) in the directory /usr/bin/firefox (the directory where you installed firefox).

you need to check the /usr/bin/ directory for the symlink you created before.
if you do a ls command in that folder, you should get something like:

lrwxrwxrwx    1 foo  foo         4 2003-02-07 16:49 firefox -> /usr/bin/firefox

rm /usr/bin/firefox

ln -s /usr/bin/firefox/firefox /usr/local/bin/firefox

The fault was found.
I wrongly forgot the line "Exec=firefox" in the firefox.desktop file.

The symlink is correct since I downloaded and installed the Firefox under /usr/lib/firefox/ according to the orders I read somewhere. Also I can activate the Firefox anywhere in the directory tree. :-)

Thank you very much Leppie, without you nothing would have been solved.

you are absolutely right.
I am sorry for my errors. The environment is new to me, which makes me confusing over and over again.
Thank you again.

you're welcome
well, we all learn (at least, most of us do) from our mistakes. there's plenty to study and learn in Debian.
why did you not use apt-get to install firefox? that's the Debian-way and much easier for newbies and it would've taken care of the main menu shortcut missing issue. about that, what desktop environment are you using?

Was not aware about this option. I am learning it from you just now.
BTW, have any idea about how to complete the FF instillation by adding it to the Gnome desktop menu?

Thanks you.

to add firefox to your menu:
- right click your applications menu and hit “Edit Menus”
- select the "Internet" section
- check the Firefox entry, otherwise click "New Item" and use "application" for "Type", "Firefox" for "Name", "firefox" for "Command" and "Web browser" for "Comment" then click ok.
- close the menu editor and check that the new firefox shortcut is under the "Internet" section