With your location, I'd concentrate on the Leghorns and Aussies. They're both more heat tolerant breeds. Doesn't mean ignore the others, but I'd set more eggs from those hens overall.
I've done pretty much what you're describing---start with a variety of hatchery birds and let them mix and match for a few generations.
Mine have all been pretty good layers. The crosses aren't going to give you something as good as a parent Leghorn or sex link, but they'll still be good...