Can I Use Polystyrene Foam Or Wood As A Core?

Yes. Yes you can.

But read this first. 

Tinfoil is always the first best option. It is absolutely waterproof, has terrific grip, is commonly available and pretty inexpensive, which is why we absolutely recommend it first. With Wood you must put on a layer or two of tinfoil and make your shapes before applying Pal Tiya Premium. This is because, as with a foam armature, Pal Tiya Premium is highly sensitive to moisture which would make it too dry, set too fast and become weak.

With foam, the short answer is paint the foam first, dry throughly, then apply foil. An outdoor Acrylic-based paint can help seal it. 

White bead foam is the exception, hot glue sticks lightly to this surface.  This would only become a problem on a very large vertical piece, such as an adult size standing human figure.  

The solution that is perfect for a project and will get the best possible results is paint first in acrylic to seal the foam, dry it throughly, then use foil and hot glue. 

Also, make sure that there is enough water/moisture present at least during your creation's first 5 days of curing. Remember, keep it wet wet wet.