I wouldn't, just out of fear that the styrofoam would leach something into the water. But, I know some folks use styrofoam underneath their gravel- for heavy rocks. Hmmm.
The 10g tub sounds fine, just set it up as a normal spawn tank.
You can slip something black underneath the bottom. I found my males had a hard time collecting eggs etc and were missing a lot till I realised that I'd got the silver shiny side of my polystyrene up and when I looked in the tank from the top could not see a thing and was blinded with the light reflecting off it
I now have some black backing under there and they never miss an egg or a fry. You could use black bin liners or anything dark really.