First thing is that you need to use a good wood glue. Staples are there to hold the wood together until the glue dries, instead of having to clamp the entire piece. They provide little to no extra strength to the joint. I used in the past Titebond II. Don't use
Gorilla Glue. That's a
polyurethane glue that expands and requires water to activate.
For frames and facing, I prefer staples. They have more surface area holding the material than nails, though I've worked with both. They simply hold the material together while the glue dries, so use whichever you're more comfortable with.
When you determine what length of fastener, I was taught that it should be no less than 2 times the
thickness of the first material you're going through, and not longer than the total length of both materials. If you are building a Hollywood
flat frame, it would be going through 1x material at .75", so a 1.5" staple would be the minimum length. You can use longer, but the longer you get, the more likely it will hit something or not be fired in straight and shoot part of it out the side of the wood. I've seen people use 1.25" staples, but they often don't hold enough in the second material to be helpful
As for the Lauan, I would use 3/4" staples. I don't think most staplers go below this length in terms of staples they can fire. Again, following the above rule, the staple shooting into a .25" piece of Lauan should be no less than 2x the
thickness, so no less than 1/2".
All the pieces of the frame (rails, stiles, toggles) should be stapled the same way with plenty of glue. You can turn the wood supporting the Lauan seam on it's
face like with a Broadway
flat to allow for more surface area to attach to, or you can double up the 1x and have one piece support each side, or run a 2x centered on seam for the same thing.
Since you are making these flats for reuse, I would suggest using a compound called Bondo Body Filler. It is a two-part putty that when dry becomes rock solid. They use it to fix car bodies. It comes in two parts, the putty and the hardener. Mix the two according to the instructions then apply to the seam, any holes left by the staples, and any Lauan
face defects, and let it dry. Sand smooth, and you can paint right over it no problem. Once you mix it, though, you have a short time period to apply it before it hardens, so only mix what you need and only when you need it. Most often for cost purposes you would use drywall spackle, but over time that can break off.