Best advice...is to do what we do. Spike aireate in the spring ( not core ) this seems to help with lawns that need just a bit of help getting that bit of energy below the root system. We do alot of these in the spring & core in the fall.

 

Yes, we seed an areate (core) in the spring and fall. Like you said fall is better. Gives the grass plants time to get established before the heat of summer.
We tell anyone that fall is better for planting. If they want new grass planted, we make it quite clear that time is running out quickly for the grass to get established and we DO NOT guarentee it to grow.