All I need is a list where subscribers can write to a generic address (like jokemail-list@myjokemail.com ) and be sent a "Confirm Letter", to which they respond and are automatically added. I was under the assumption that Mailman can do that stuff.

Instead, all I see is a program where my subscribers would have to go to a separate webpage that Mailman provides, and submit their address there.
What's worse is that they even have to have a password. This is nothing like I though this program would be like.

Of course you can subscribe via email; a message to <list>-request with "help" in the subject is probably the easiest way to get instructions on how to do that. See templates/help.txt for what that message would contain, if you haven't yet installed Mailman.

Another thing: having 5000 people, isn't there an easy way to transfer all of them over to a file, instead of pasting their names in batches of 30 addresses to the configuration page?

When you say "transfer all of them over to a file", do you mean "transfer them to be subscribers to the Mailman mailing list by putting their names in a file and then submitting that file as input to Mailman"?

The "mass-subscribe" option on the Membership Management page will take more than 30 at a time, so that's not a limitation you need to worry about as far as I know...but easier than that is to use ~mailman/bin/add_members. Read the top part of that file for documentation.

So I guess I am writing this for advice. Does this program do any of the above things that I am looking for? I dont need my subscribers to have a password...simply a letter to confirm their subscription request upon submission.

Mailman requires that your users have passwords; it's simply part of the program. Many people have asked for this to be configurable, but it is not yet. But the password can be randomly assigned, and can be changed by the user through the web page at any time.