Foam Selection: I would stay away from expanded polystyrene (white bead board) because of the lack of strength. That leaves extruded polystyrene (blue or pink) foam board, or more expensive types. If you go with pink, its probably Foamular. Some of its properties are:
Type Comp Str. Flexural Str. Density (lb/ft^3)
150 15psi min 60psi min 1.4
250 25psi min. 75psi min. 1.8
400 40psi min. 115psi min. 2.04
Nowhere near the Corecell that John mentioned: (density of type a, 3.6lb/ft^3), then again... we're not trying to hold up a house. If you can get the Foamular 250, its higher strength is better. Its typically used under concrete for foundation insulation. Don't know if HD has it, but I've seen it at Menards, our reginal big box. Closest Menards to you is probably Jackson, MI. Most lumber yards should have, or be able to get the 250 for you. Last time I looked, it was $30/sheet vs $15/sheet for Foamular 150.
Glue selection: I used polyurethane glue to attach my insulation to my sides. I like its gap filling properties. I actually misted the side sheet before applying to induce the foaming. Elmers brand is less viscous than Gorila, and tends to foam more. I used a water based multi-purpose tile adhesive for my floor foam. I don't know how well its bonded, but I'm not really using it structurally.
As GPW said: the strength is in the skins. As long as the bond between the skins and foam is adequate, and the foam can take the stress, its good to go.
How are you going to use these panels? Standard SIP construction (one method) is to assemble the panels, cut the opening, gouge out the foam around the openings, and put blocking in. I think its easier to put the blocking and foam on, then apply the second skin.
Captainsam: check out
U. S. Composites 80cu ft kit for $209 2lb/cu ft densit. Also, for your 4'x8' panel: I was think of how to clamp it, and came up with an idea using a self-propelled weight. Find a flat spot in the driveway, build a frame out of 2x10's and deck it with 3/4 plywood, put side one down, pour the foam, place second skin and deck on top, and drive the front of a car or truck on top of the frame. Kind of silly, but I've been thinking about this for a while. (my foam kit should be arriving today, so I get to play with the stuff

)