Nothing is impossible. Except this.
In standard HTML or JavaScript, you can't make raw-socket connections, which is what you'd need to speak any of the protocols you've listed. The WebSocket API doesn't help you, because the server must also speak WebSockets, and mail servers don't. This is actually all a good thing, though. Imagine if you visited a random website and it telnetted to your home router, setting it on fire and burning down your house. That's what websites would be able to do if they could initiate plain TCP socket connections.
You have a few options I can think of, neither of which involves building a webpage. (And to be clear, you didn't say your project had to be a webpage; you merely said HTML/JS, so these are indeed legitimate options.)
First is something like node-webkit. As the readme says, "You can write native apps in HTML and Javascript with node-webkit." You'll still have to distribute it as a native app, because that's what it is.
Second is a Chrome app, specifically using the chrome.socket API. But have fun writing a TLS layer over those sockets, which you'll find is a requirement for almost any mail server these days. If you succeed in doing that, you'll be able to distribute your mail client in the Chrome Web Store, where I assure you an email client will be quite popular.
Third, write a webserver that operates locally, so your users will visit something like http://localhost:9999/mail
in their browser. This will be a real pain to distribute, but you can use almost any technology you want.
I'm sure there's a fourth valid option, and someone else can chime in on that one.
Best of luck.