FrankenShop, what I would do is buy a prebuilt wood screen door for your opening which I'm assuming is 32"or 36" wide. Once you get home add 1-2" or so on to each stile to increase the width. It's not structurally going to be supporting anything, it's just going to be overlapping your trim, so exterior wood glue and clamps is all you should need. Your height should be okay but you may have to add on there as well, but again it's not structural as your roller assembly will be fastened to your top rail. Your door will already have screen in it so build a frame for your dog door as shown in your diagram and add it to your door (pockets screws and exterior glue?) and staple the screen to it. Cut the screen inside the opening. Install the dog door and the trim on the dog door will cover the edges of the screen and your staples. Add some sort of brush style weather stripping to your door trim to help keep out "most" of the bugs.