The valance is 42". Does it need a strip of wood on the back top edge for support and prevent warping?

Instead, would making a box (wood@ceiling) be necessary with the arch being the front of the box?

Pine

You could add a strip at the back top edge, as you suggested, and then attach the strip to the ceiling. Do the side cabinets meet the ceiling? If not, then the valance by itself might look a little weird if you can see the open space above/behind it.

My preferred method would be to attach the valance itself to the side cabinets using pocket screws on the back side (pre-drill the pocket holes before installing, then screw in place).

Pocket hole screws work well but not worth buying a jig just for this one time use. Here's an easy and inexpensive way to do it,assuming valance is 3/4trs" thick. Using 2 people,hold valance in position then draw a light line along each end where end of valance contact's cabinet. Lay valance aside and drill two 1/16th" holes 3/8ths" from both lines drawn earlier. Hold valance even with lines and drive finishing nails through holes from inside cabinet and into ends of valance. If you think you might want to remove valance later on,use screws instead of nails.