You can't really use contact cement or hot glue on Styrofoam ("Styrofoam" is just a brand name, what you're using is actually called Extruded Polystyrene or EPS Foam) or any kind of polystyrene because it will melt (sometimes low temp hot glue will sort of work). Certain adhesives are designed to work on Polystyrene; I personally use Weldbond, but 3M77 or any Styrofoam glue should do the trick.
If you sand the back, edges and post of the half dome eyes Weldbond should hold everything together nicely (just give it lots of time to dry). I would suggest trying to find Polystyrene balls with a smoother surface because they will be much, much easier to glue anything to.
EVA foam balls - which are often sold as yellow foam practice golf balls - would be a good Polystyrene substitute if you can find them in the right size. You can use almost any adhesive on them (hot glue, contact cement, Weldbond, etc.).
2" plastic spheres are available online if you do some Googling (not sure about cost though). They are often clear, but you just have to sand them and paint them. Or why not use a 2" fishing float?