Thank you for your detailed answer.
Now it seems it is working. I think that Thunderbird see only the messages that has been received by gmail AFTER I have installed thunderbird on my computer. It is strange, but now it react at all the received messages.
I will try to work tomorrow at office to try to set my thunderbird under a proxy - I tried what was indicated usually on the forums
but I did not succeed. Maybe I have done something wrong. I will keep you online with my attempts.
Thank you again !