If it was tilled last year, don't worry about tilling again. Use a pulverizer to break up the crust, then seed. I'm not a big fan of spreading topsoil for a new lawn, it really doesn't do much, and probably isn't very cost effective in the long run.
If they want a "carpet" of grass by August, it's do-able. Make sure it's watered correctly, and it can be done.
If the trees are large and cast a lot of shade, make sure the seed mix contains red fescue, it does the best in shade. Sunny areas, use an 80/20 or a 70/30 mix of bluegrass/ryegrass.
When you seed, use a starter fertilizer. Or just put down Penn Mulch and forget about straw. PM contains a starter fertilizer. We've used EnCap Mulch, but will probably go back to Penn Mulch in the future.