Have the same setup in my master bath. Ignore peanut gallery comments.
I was able to fish a line from under the vanity to the TV location. I you are already commited to drywall repair, a few more small holes won't matter.
I am assuming you are going to have to install another electric outlet for the TV location, while you are doing that, loop it and the satellite feed to a cabinet under the vanity or a nearby closet with an open shelf, also install a component video or ( if price is no object) a long HDMI cable from the receiver location to TV mounting location. Remote control offers two options. Purchase the RF remote kit for your receiver if it didn't come with one, or install an IR repeater with a plasma or compact floresecent light (CFL) friendly IR receiver mounted by the TV. normally can use phone wire, microphone wire, almost any small guage 3 conducter wire to run between the receiver and TV. Xantech is one manufacturer, but there are many others. If your are dead set on mounting the TV in such a way that the display is flush with the wall, I would definitly think about making sure there is no insulation in that cavity and would cut out an opening at the top of the cavity near the ceiling where you could mount a cold air return vent cover to allow heat a place to dissipate but still not look too cheesy.