Well I've tried the Webmail for Hotmail, and indeed it doesn't work anymore (it did a few months ago) but I've found another app that can download emails from Hotmail : FreePOPs
All you have to do is run it, create an account in Thunderbird, set the port to 2000.
It's not perfect, because AFAIK you can't send mail from Thunderbird with it, and it downloaded only the first page of emails in my Hotmail inbox (and didn't delete them).
But that's fine for me because it's making my move to Gmail much easier (which, BTW, supports forwarding and fetching, in case you want to move in the future)...
EDIT: unfortunately since it only downloads the first inbox page, I had to move my sent messages to my inbox, and it downloaded all but the last 3 emails, and it generated an error message in my Thunderbird inbox. No idea why it didn't work for the last 3 (trying them one by one doesn't work either).