When I switch between tabs with alt+tab, I don't want to see thunderbird. Is it possible?
Install MinimizeToTray revived in thunderbird (you can do this via add-on menu inside thunderbird also).
Once installed, go to its preferences, and select the option "instead of closing and when minimizing" in the "Minimize to Tray" section.
Now after opening thunderbird, you can close it from the "close" button in titlebar, and it will still keep on running in the background, checking mails periodically. But it will no longer be an open "window", and hence wont appear in alt+tab.