The kids go out even when it's 0°F and snowy. There were a couple of days at below -10°F when they did not go out, and a few days when they only stayed out for maybe an hour instead of 3.5 hr. In the winter, they wear snow pants, heavy jackets, gloves, hats, face mask/scarf. In the rain they wear rain jackets, rain pants, and rain boots. They do just fine. The teacher for my kid's class warns ahead of time if they expect different gear to be needed than normal for the season.
My son's class of 14 this past year had a teacher, a naturalist, and two aides. This is a public school system. I do not know if private classes would have more or fewer adults.
So, fights with sticks could happen, but they don't seem to any more than any kind of fight in an ordinary classroom might, and the teacher/aids deal with them the same way they would with classroom fights.
When the kids are out in the woods and not in the outdoor play areas, they are more closely supervised. They don't have free roam of the entire forest. Hiking and adventuring is led by the naturalist and assisted by the teacher and aides. They are taught about things like "don't eat the mushrooms" and such. Still, things happen. My kid loved to jump into water, and he always got his shoes/boots/etc wet, even when it was cold. One time he broke through the ice at the very edge of the lake (deep enough to get into his boot). The adults and kids deal with it, just like we had to in scouting when I was a kid. The main difference is that when I was a kid, I only got to do it about once per month on a weekend, whereas my kid could do it all day every (school) day. How awesome is that?
When I was 14 or 15, I slept in a tent in the deep woods in -20°F overnight. It was really no big deal. My kid at 5 is already at least as good of a naturalist as I was at that age. :)