Health: I think you’re making an unfair judgment about someone you’ve never met, but it is probably a true judgment to make of most women who would wear Tzitzis out.
#1: I don’t think so. B’nei Yisrael V’lo B’nos Yisrael doesn’t create an Issur. It’s explaining who the Chiyuv is on. And since women aren’t Mechuyav in Tzitzis, they are not capable in creating a Lishmah for making the Tzitzis. That doesn’t mean they aren’t M’kayim a Mitzvah K’einah Metuvah V’osah if they do it though.
#2: You’re right. According to the Mishnah B’rurah, nowadays when everyone (especially women) are much more Makpid on hygiene there is no reason to be Mocheh a woman who wears Tefillin L’Sheim Mitzvah (according to my M’halach that I got from my Rebbe’im, however, there still would be).
#3: A Machlokes Haposkim doesn’t inherently create a Safek. It could just be that her Rov Paskens like those who hold that it’s not Begged Ish.