It's like Pandora's Box, so be warned.
The Big Thrifty Thread is the genesis of "The Modern Foamie Movement" (my pet phrase). It is a monster thread with a lot of OT, but also a lot of development work and conceptualization.
It comes with an
index thread (a work in progress courtesy of Rowerwet), sort of thrifty shorthand.
In a nut shell you have white bead expanded polystyrene (EPS) foam which is the generic version of the trade named "Styrofoam".
Then you have the pink and blue (sometimes green) construction board that is extruded polystyrene (XPS). Some of the thinner stock sizes have a thin clear film on one side (for handling integrity) that must be removed (peeled or sanded off) because it does not take glue. It is reasonably available in 1/2, 3/4, 1, 1-1/2 and 2 inch thicknesses, in 4x8, 2x8 and fan fold sheets. Some are tongue and groove, and there are also "scoreply" versions that have precut slits nearly thru (that's what I ended up with and other than causing some variation in surface level at the scores, has not been an issue).
Solvent based contact cement will eat the foam; but you can use water based, such as 3M 30NF Green (expensive).
Fiberglass (taken to mean polyester resin, Bondo, etc.) will eat the foam. Glass reinforced epoxy works great, so long as you do not use a heavy/hot batch; high heat will melt styrene.
Gorilla Glue (GG) is best for butt joints and will also work for smaller laminating tasks, but requires attention to locating, clamping, and masking due to expansion.
Great Stuff (GS) works well for filling and repairs, and at least one user reports that it works good for laminating and butt joints if you knock it down to prevent foaming. Clean the applicator tube with a drop or two of acetone; melts foam clean.
My own experimentation has proven that TB2 works good for wood to foam if the joints are tight. GG is better for so so fits.
TB2 also works great for laminating wood skins to foam provided you apply the glue evenly with a roller. Avoid puddles, or too much glue; the wood needs to be able to absorb all of the moisture in the glue since the foam will not allow it to air dry.
Personally, I did the exact opposite to what you are proposing. I made my walls and floor out of foam with thin wooden skins and will use a piece of well supported 1/2 inch ply for the structural deck of my counter. Contemplating a thin SS sheet for the finished surface. Durable and will take the heat from a heavy DO.