The drywall may provide bracing along the studs, but a groove will not impair that. However, you will be creating a discontinuity, and if there is any stress on the drywall in the future, due to movement of the foundation, or shifting or shrinkage of wood members, you will get a crack along the weakness. This is why good drywallers don't extend seams from the corners of windows and doors.
There should be no reason you cannot simply put a pair of holes in the drywall near the top and at the bottom and fish the wire. There may be fireblocking in the wall, about mid-height. Across a ceiling, I'm guessing you don't have an accessible attic space. If you have crown molding, there is probably a gap behind the molding. Drill a pair of holes where you need them, then vacuum a string with a cotton ball tied in the end from one hole to the other (works if the drywall is properly closed.) Pull your wire through with the string. Another thing I have done is to add a small molding to the edge of an existing one. The slight change in profile adds to the look.