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You are welcome, believe it or not the biggest challenge is gluing the foam to itself and to the wood sides.For that I have discovered Glidden Gripper (now by PPG - so called PPG gripper) Primer. It is a latex based paint that has adhesive in it. it still dries slow when no air can get to it, but it works as a combo paint, primer, glue and will glue the wood to the foam and the foam to itself.
Warping not really been an issue if you are layering the foam board. Contact adhesive should work as long as it is not solvent based.
I would agree, I only use it on the edges, not under.
1/8" or 3mm depending on where you are. I like Luan because it is light and relatively inexpensive, Baltic birch ply available in 3mm is also a good choice.I don't use MDF typically because of weight, plus many versions of MDF are more susceptible to warp.If used on the side edges it helps keep thing straight and minimizes chips in the foam. Then with some well placed dowels and holes, you can keep things lined up when you slide the sections together. Add the over center clamps and the whole thing becomes very stable.