If I download e-mails from my account using a client like Thunderbird, can I just somehow copy them to a thumb drive and open them in whatever program later, without logging in to the account etc?
I want to delete my account, but need access to all my e-mails off.
Yes and its very easy actually.
They are just specially formatted text files that can be opened with email programs or text editors. This means you can easily do a full file content text search through the entire folder if you need to find something.