Tv remote connection to an Arduino
If you're just running lines to the buttons, you won't need to power the remote from the arduino.  You could just let it keep the internal batteries.  But it would be possible to wire it up for power as well.

As a learning project, you could just take it apart and see if you can access the button contacts.  See if the contacts are split.  If so, shorting one side to the other should count as a button press.  But the details have nothing to do with an arduino.  They would depend on your specific remote and how it's constructed.

If your purpose is really to control the TV (rather than learning about the remote and fiddling with it), it will be much easier (and probably cheaper) to blast the IR without involving the remote at all.  A single IR transmitter and a library would work in most cases.  As an example: http://www.righto.com/2009/08/multi-protocol-infrared-remote-library.html

P.S.  By coincidence, Adafruit just posted an updated guide to IR library use on Arduino.  https://learn.adafruit.com/using-an-infrared-library/overview