I just done my whole back garden,it had a load of pebbles and railway sleepers through the middle of it so I ripped it all up,moved back some of the flower beds and put grass down in the middle,the grass area is about 12ftx20ft
In all I used 4 ton of topsoil,I got the screened soil,so there wasn't that many stones in it just a few pebbles.It was back breaking enough having to bring the soil from the front of the house to the back but I got the 4 ton in,raked and rolled in a day.
I too wouldn't bother with the turf,if you throw a few seeds down it will grow in no time in this weather,the mixture of heat and a bit of rain and you should see signs of grass within a week I reckon