I don't think alarming emails are sent. Many of us, like you, rarely check our email, so the alarming feeling when they're sent is lost when we actually check it. It's something to be suspicious of, too. In fact, I find it absurd. I completely agree with you.
You are so wrong. Plenty of people check their email just like their phone every few minutes. How important can it be? Or should I say: can one be.
I don't check my email constantly , in fact, I do it only occasionally, and there's almost never anything new. But that's perfectly respectable, everyone is free to do what they want.