complete/underview status will depend from school to school. some schools will separate them, some won't. some schools will go straight to under review..or change your status from complete to under review. some will leave you on complete and never change it until invite interview. Really, what I meant by "complete/ under review" is just when i turned everything in and when the schools officially acknowledged that. also, my advice...call each school after you have everything in. some schools will check and some won't *cough cough* UPENN.
I would go by the emails from each school since they email and give you updates. So each school will send you an email when you have EVERYTHING in and usually the email will say something like "you're now complete/ you're now under review/ you have everything in and we don't need anything as of now, unless stated in the future." that's when you know you're have done everything you can and the waiting game begins! it's painful.
but to the question of the post, i don't think end of july is late. however, if you have lower stats...I would say to try to turn everything in sooner just to be on the safe side 🙂 also, keep in mind that once you take your DAT, it takes about 3-4 wks for your score to be go from "unofficial" to "official" for the schools. and schools want official scores.