Do I need to install flashing for this replacement window?
The purpose of the flashing it to direct any water that manages to get behind the trim back to the outside surface of the house instead of letting it continue to worm its way inside where it can start to rot the framing.
The pieces of wood you've identified don't have direct paths to behind the siding, so they don't need to be flashed.
Keep in mind that the trim, framing, and windows themselves have been made of wood for many, many years. There is no need to apply flashing to these bits of exposed wood.
It sounds like you've got a good plan of attack for preparing and protecting the wood, just keep an eye on it (a yearly check would be good) and redo the scraping and paint job as necessary.
From the looks of it, you let it go too long since the last repaint, so don't let it get this far the next time. Don't worry, I'm not taking you to task for that - you window looks far better than mine did before we finally got them all replaced.