Can a proxy make me anonymous to HotMail, Yahoo! Mail, etc, so that I can send anonymous e-mail?

Yes. As long as you are connected to the HotMail server through an anonymous proxy server, your IP address is hidden, and all e-mail that you send with the web-forms on the HotMail server has no track of your true location. It is so for all web-based e-mail services, like mail.com, as well as for web-based chat boards like Yahoo! Chat. There is a possible trap, however: many of such services use a secure connection when you send e-mail or when you logon to your account. Therefore it is very important to make sure that your proxy supports HTTPS and configure your browser to use the proxy for the Secure protocol in addition to the ordinary HTTP.