In the first case, you must take action to appease them, whether replying to the email or taking their call. (Or you could script your interaction with them; an automatic email reply or such. But that's suboptimal too). It's outside of your control.

In the second case, they aren't able to distinguish what they consider to be emergencies from what are actual emergencies; this, too, is out of your control.

I honestly don't know how to deal with these. Thoughts?