4-5" is
just enough, IMO. Wood chips or saw dust in top soil mix tends to fix the available nitrogen in the soil making it unavailable for the new grass, so it's a good thing you didn't use it.
Rent a roller.
Peat moss, plenty of dolomite lime and granular gypsum will help with clay soil. Too much organic material in the soil and it'll retain water like my sisters...well, nevermind.
I've got a black thumb when it comes to growing that other grass.