Yes, you can, you have a user profile in Thunderbird, that is where your emails are stored locally.
You should first look up downloading your emails locally, this depends on the connection to the mail server.
If you are running POP3, it normally downloads your emails directly to your computer automatically, if you are using IMAP, then it normally doesn’t.
Thunderbird is a popular email client, there are plenty of guides…
Nope, that is the speedycake remix of Caramelldansen