I would install a drip edge on the top frame so that the water coming down the siding will channel out and over the wood top frame. The rest I would let it alone. If you box it in and water gets in it will stay, but if un caulked it will air out and drain out. Caulking lap siding with a framing over it is difficult to do.
I would definately keep it painted. Is the wood pine or what for the frame?