(Not a Teacher. BrE first language speaker)
The only thing I might add is that to me "cool it" feels like a step up from telling someone to "simmer down", but a step below telling someone to "shut up".
I can't think of the last time I have said, or had to say, "simmer down" to anyone and agree with Barb_D it often has the opposite effect when used on other adults. To me it feels like the preserve of those that need to keep control such as teachers or the police.