@bookmom7 My son is applying for PhD in Math! 22 schools!!! - His supervisor told him to expand his list of “safeties” by a lot). In fact, my son managed to get through all of those applications, except for one, which requires the residency proofs or whatnot by parents.
The useful tidbits for you: first off all, there are a great discussions in the forum https://mathematicsgre.com/ , where previous applicants posted the list of schools they applied to, their GPAs and GRE scores, and whether or not they did research, and whether they had publications etc. And then, these people update their posts to reflect the schools that admitted them, with funding or not, etc. This is a great place to sorta “scale” yourself against other applicants, and use as a source to reflect on your choices of schools to apply to.
Also, my observations of my son’s progress told me that it makes sense to initiate all applications first, so that the requests for letters of recommendation can be initiated ASAP - because some schools require those letters to come before the deadline. When applying at the due date, if the rec letters were not sent earlier (before the application was submitted) - it is then impossible to “make” the deadline… (Just from one mom to another…)