My sons go to a school that sounds very much like the one your DD is attending. A secular private school, where everybody is addressed by their first name (students, teachers, staff, administration.) No uniforms. Pretty progressive in their teaching methods -- they aren't mired in the 1960s. Funny thing. Almost everyone goes on to post-secondary education of some form. Many end up at US Ivy League schools, or places like Northwestern, Stanford, Cal, USC, UCLA.
Suggested respons: "MIL, we've made the best decision for our DD. We know the difference between OurSchool and the one that niece and nephew attended. Here's the literature on OurSchool; please read it." After that "The subject is closed. Have you tried the Vegemite dip?" Then don't respond at all, but leave the conversation.
As for the other stuff, she's being a "Concern Troll," acting as if she had your best interest at heart, but really undermining everything you do or say. There are a couple of techniques for dealing with a PA person like this. One is the "what did you mean?" method, of making them explain their PA remark in excruciating detail. The other is to take what they say absolutely literally and respond to that. I think the former is better based on the type of comments you mention in the OP, but keep the latter in your arsenal as well.
"Interesting assumption" can also be used, but I don't think that it's as effective with PA people. She sounds like she knows what she's doing, while I think "IA" should be reserved for people who have made a bad assumption.