First you don't want to secure all sides of the panel or it'll break the door or split the panel. Solid wood needs to move.
Secondly you may need to touch up the panels during drier times as wood shrinks when it dries.
I'd secure the center of the panel on the inside of the door with some super glue If you have spray activator it'll make quick work of securing the panels. A little dab in the center top and bottom and it should stop the panel from sliding back and forth. The only modification to this is if the grain is running parallel to the floor and not up and down. If the grain is running parallel you would secure the sides.
Post some pictures if you are unsure.
Absolutely sanding is required, rasps do not leave a smooth surface. But what rasps do well is shape and blend the surface, taking the irregularities out of the surface so sanding can can result in a smooth surface.
Cheap rasps will work, but the nice ones are a lot better. They leave a relatively smooth surface, esp with the less coarse rasps. I never thought I'd ever use rasps as much as I do now, as I used them in projects I just keep finding new ways to use them.
I have my cabinet doors done and ready to paint but I've noticed that the floating panels have some movement that may show unpainted sections if they shift. Any recommendations for fixing a panel in place after its been assembled? Simple shaker rail and stile and for most of the doors the fit is tight enough that they don't move but there are a couple that can slide.