It sound strange tihat you would be blocked because of you keeping your TB running. Check that your mailserver-setup is right. Check with your ISP if they changed anything. Webmail is a total different way of connecting so my thoughts is about setup.

i received a message from google they thought it was an unauthorized access because I was connected via portable from office... only tb was open and now seems permanently blocked

login to your account on google and look in your personal settings. I have a Gmail-account and there is a place under account that say restore-mail that might be the answer. search google.

And if you change password be shure to change it on TB to

I was on my gmail account and looked at settings forward and pop/imap be sure they are activated. Imap if you want to access it from several places. That way you have the original on Gmail-server and copies on your pc. Pop on the other hand loads down the emails. perfect if its just one pc. (i'm not sure the Gmail-server deletes them thou).

Just a thought:

Gmail changed it settings and uses a new security on new accounts.

login to https://www.google.com/settings/security on your account and allow "less secure apps"

And its NOT unsafe Tbird uses normal IMAP security.

good luck