I'm sorry - kids do not get sick from being in the cold weather. Neither do adults. That is a huge old folk tale and it's just not true. Short of frostbite and exposure over a period of many hours, there's no risk. The reason we are sick in the winter is that we don't get enough fresh air, and we are indoors in dry, heated air where bacteria and germs breathe.
I can see being a little concerned about shoes being trashed - boots might have helped that. The school should have a policy that every child bring boots every day if that is a concern of parents.
Perhaps the reason the teacher looked at you that way is that you have previously complained about them NOT going out - now they go out, and you are unhappy.
I am quite sure that the state's office for children would not say that kids should be indoors if it's 43, or even 32. If your daughter was sweaty, she was already plenty warm from running around. If her hands were dangerously cold, I hope she would have said something to the teacher - but if she wasn't uncomfortable, what's the problem? I mean, exactly what temperature would meet with every parent's approval?
I was always thrilled when my son was outside running around since we have such a problem with kids being sedentary and sitting in front of video games and computers. I believe in sending kids to school in play clothes that can get dirty and wet, and letting them have fun and explore! I never sent him to school in good stuff.