If you are going to paint the repair, use drywall mud. It is shapeable and easily sanded. You will need to allow extra drying time if the repair thickness is more than 1/4"---and you may need to sand and add more after the first application.

Just primer over the patches and then primer the entire piece.

I have seen several DIY shows that recommend using Bondo. Yep the stuff used for cars, to repair wood window frames and sills. Just passing this along, as I have not tried it.

Maid

For the filling, obviously a slip of wood is the craftier solution.
If using bondo (or the minwax stuff) I like to work the patch with an overlay of wax paper, which allows one to get a smooth fill without pulling at the edges. You peel off the wax paper as soon as the bondo begins to set; you want to use the cheese grater/Surform tool to level the patch before the bondo gets ultimately hard, which it will do within in an hour. After which time you will have to use a wood sanding block to level it.
Casey; Bondo's in my veins.

I have repaired numerous widow sashes and door casings with Minwax.

You can easily shape it with sharp wood chisels and carving chisels before is sets up completely, then go back and clean it up again with chisels and sandpaper once it is fully set.

Bondo is harder once it sets and a little tougher to sand out.

Thanks much for all the suggestions.

"Bondo is harder once it sets and a little tougher to sand out."

That is a very kind statement indeed

I agree with sombreuil_mongrel. Moreover I agree with your ideas. Thanks for sharing.