While styrofoam insulation will help prevent sound leakage, it won't improve the room acoustics much. You need to make your room less reflective to sound. Anything absorptive (bass traps, acoustic panels, filled bookshelves, fluffy sofas, etc.) will help with that. This will help with the first of the two problems you mentioned.
As for the rattling radiator, you probably need to figure out where the rattle is and fasten it down or something similar.