Can I setup my Hotmail to automatically forward a message from a single sender (not all messages coming into my Hotmail account) to another email account (like Yahoo or Gmail) and then have the message automatically removed in Hotmail?
While the new look of Outlook.com is awesome, some of these types of advanced features are simply not yet available. Currently Outlook.com only supports "basic" Exchange rules and therefore only support one action. Once a rule matches and executes, the remaining rules are ignored as well, so there is no way to bypass this limitation.
Obviously you can enable a forwarding rule, but you would have to delete the email from your inbox manually or employed client rules in the Outlook desktop app.
Here is a recent Reddit post from Microsoft indicating that they are aware of this and working to improve things: