Sending an email is something totally different again - you can send yourself or anyone else an email to any email address.
You can also send or forward any email while on your Yahoo account to yourself or anyone else.
Setting up Yahoo to automatically forward emails to your home email address is something else. Here is an excerpt from Yahoo Help:
+The ability to access Yahoo! Mail using an email reader program such as Outlook, Eudora, or Thunderbird is only available to customers of our premium Yahoo! Mail Plus service. To determine whether you have purchased this service, please visit the My Services page.+
Although this does not mention Apple Mail, it functions exactly like Outlook, etc. so it will not work. Here is the link to the entire help page:
http://help.yahoo.com/l/us/yahoo/mail/yahoomail/mailplus/pop/pop-35.html;_ylt=An EHIP08O6GRC7cEYzjbTbe7myN4
I have no idea why it might have worked for you previously, but according to Yahoo, it cannot be done unless you pay for the service.
So, unless I've totally misunderstood what you are trying to say, I do not have a different answer and it does not have anything to do with your Mac or OS.