I'm not familiar with Thunderbird. But, I believe getting all the symptoms is half the cure.
When you say "after several days it does not appear", does that mean that you can see the email initially, but it later disappears?
Is it just your Sent folder, or does other mail disappear as well?
Are you accessing your GMail account from any other programs, accounts or apps? Possibly one of these is accessing your account via POP?
In your settings for POP download, do you "keep Gmail's copy"?
https://mail.google.com/mail/u/0/#settings/fwdandpop