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The HDMI connection provides 5V power.

I believe that is only true for MHL HDMI. They likely use USB power so that it can be used on any HDMI port, not just MHL-enabled ports.

HDMI does provide power [0], but a very small amount (elsewhere on this thread someone said 50mA). MHL provides more.

I don't know what MHL HDMI is. How do I verify if I have one? My TV is 2009 Sony Bravia and it transmits audio (since I can use Apple TV).

I would guess that you don't have it. The Roku Streaming Stick uses MHL and it only lists a handful of compatible TVs. http://www.roku.com/streamingstick#roku-ready