If I cut the groove shown in the drawing when the 12 foot PVC board is flat, will the groove maintain it's dimensions if I then bend the board into an 8 foot diameter capital C shaped semi circle?
IMO, to maintain the shape of the groove while bending the board, one may need to have a good control of the heating method if any in the bending process and some sort of guides needed to keep the dimensions as wanted.
It's going to change some, probably not much. If you need a lot of precision take a small section with with a groove, bend it to the same radius, and verify results. If you're going into mass production you may consider shaping your groove so that it deforms into the correct shape.
Yes, you risk distortion along the edges of the bottom of the groove considering that its depth is greater than a third of the thickness of the plate. In this case, the likelihood of 'plastic' (no pun intended) deformation is very high.
Fairly easy to route or dado the slot after bending, unless this is on the inside of the bend. If so, make a piece to fit into the groove as the part is bent.